WEBVTT mathematics/ap-calculus-ab/hovasapian
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Hello, and welcome back to www.educator.com, welcome back to AP Calculus.
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Today, we are going to talk about the maximum and minimum values of a function.
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A function on a given domain, it is going to achieve different types of max and min.
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We are going to talk about an absolute max, an absolute min, local max, and local min.
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Let us jump right on in.
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Let us start off by just looking at this graph.
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Let me go ahead and tell you what this graph is.
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It is what the function is actually, that this graph represents.
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Let us go to black here.
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Let us call it f(x).
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Here f(x) is equal to 3x⁴ - 14x³ + 15x².
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The domain is restricted on this one.
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The domain happens to be -0.5.
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X runs from -0.5 and it is less than or equal to 3.5.
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We know that a function is complete, when you actually specify its domain.
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Generally, for the most part, we do not talk about domains but domains really are important.
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In this case, even though this function is defined over the entire real line, we are going to restrict its domain here.
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Notice that this is less than or equal to.
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The endpoints do matter, so it is defined.
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We have a point there and a point there.
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Like I said, we are going to be talking about an absolute max, an absolute min, a local max, and local min.
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I’m going to go through a few of these graphs and talk about it informally.
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And then, I will go ahead and give formal definitions for what they are.
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The formal definitions are there just for you, they are in your book.
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Our discussion right now in the first couple of minutes is going to make clear exactly what these things are.
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The absolute max of a function on a given domain is exactly what it sounds like,
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it is the highest value that f(x) takes on that domain.
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In this particular one, this point right here, which happens to be 3.5 and 33.69.
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This is the absolute max.
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On the domain, it is the highest value that f(x) actually achieves.
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We know that if did not restrict the domain, it would go up into infinity.
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In that case, there is no absolute max, there is no upper limit that we can say.
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But here, we can because we have restricted the domain.
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Let us talk about the absolute minimum.
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The absolute minimum, the lowest value that a function actually takes on its domain happens to be over here.
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This point is 2.5 and -7.81.
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This is an absolute min, it is the lowest value that it takes on its domain.
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Let us talk about something called a local max and local min.
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A local max and a local min, that is where the function achieves a local max and local min at a point x in the domain.
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Such that, if you take some little interval around that particular x, that f(x) is bigger than every other number.
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Or the f(x) is smaller than every other number.
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In this case, this right here, this point which happens to be the point 1,4, this is a local max.
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It is a local max because if I move away from the point 1, if I move a little bit this way or a little bit this way,
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notice the function is lower than this, the function is lower than this.
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For a nice small little region around the point, if at that point the function achieves the maximum value that it can,
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in that little bit, it is a maximum locally speaking.
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Locally meaning some little neighborhood around that point.
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Here this is an absolute max, it is also called the global max.
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Absolute min also called the global min.
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Overall, what is the biggest?
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Locally, that is this.
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This is a local max.
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Interestingly enough, this point right here also happens to be a local min.
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This point 2.5, if I take a little neighborhood around 2.5,
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within that little neighborhood locally around the 2.5, this is the lowest point.
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Because if I move to the right, the function is bigger.
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If I move to the left, the y value of the function is bigger.
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This also happens to be a local min.
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That can happen, a local min can be an absolute min.
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A local max can be an absolute max.
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You can have more than one local min and local max.
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You can have only one absolute max and absolute min.
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You might have no absolute max, no absolute min, but you have a bunch of local max and min.
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You might have no local max and min but you might have an absolute max and absolute min.
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These all kind of combinations.
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In this particular case, we have an absolute maximum that it achieves.
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We have an absolute minimum, also happens to be local minimum.
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We have a local maximum.
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This end point over here, it does not really matter.
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It is the y value is someplace in between.
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That is it, that is what is going on with absolute max, absolutely min, local max, and local min.
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We also speak about the point at which the function achieves its absolute max, local max.
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In this case, this function achieves an absolute max at x = 3.5.
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It achieves an absolute min at a local min at x = 2.5.
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It achieves a local max at 1.
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The values themselves are the y values, 33.694 and -7.81.
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Let us look at another function here.
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Once again, a graph can have all or none of these things.
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Let me go ahead and write that.
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I think I will use blue.
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A graph can have all, some, or none of absolute max, absolute min, and the local max and min.
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In this particular case, let me see, what function I have got here, it looks like the x² function.
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Here we are looking at the function y = x².
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Once again, we have restricted its domain.
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2x being greater than or equal to 0.
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This point is absolutely included and this just goes off to positive infinity.
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In this particular case, is there a highest point on this domain?
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No, because this goes up into infinity, we cannot say that there is an absolute max.
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There is no absolute max.
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Is there an absolute min?
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Yes, there is, this is the lowest point overall on this domain.
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It is lowest y value.
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This point 0,0 is the absolute min.
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Are there local max or min?
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No, there are not.
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There is no local max, there is no local min.
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You might think yourself, could this not be considered a local minimum?
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No, a local minimum requires that at a point, in this particular case 0,0,
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that there is actually an interval to the left and to the right of it, which satisfies the conditions.
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In other words, yes, if I move to the right, it is true.
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The function gets higher in value but it is not defined to the left.
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It is not defined to the left but for a local, I need something that is defined around.
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An interval around it has to surround that thing.
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The local min always looks like a little valley.
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A local max always looks like a crest of the hill.
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That is it, that is local max and local min.
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In this case, absolute min and no absolute max, no local max, no local min.
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Let us take a look at another function here.
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This particular function right here, we have not restricted the domain at all.
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This goes off to positive infinity, this goes down to negative infinity.
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It looks like some sort of a cubic function.
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I actually did not write down what function this is but that is not a problem.
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We are here to identify graphically absolute max and min, and local max and min.
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In this particular case, there is no absolute max and there is no absolute min.
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However, we do have a local min and we do have a local max.
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Yes, there is a local minimum and it looks like it achieves that minimum at x = 2.
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There is a local maximum and it looks like it achieves that maximum at x = -2.
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The maximum values and the minimum values happen to be the y values.
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Whatever that happens to be, it looks like somewhere around 16, something like that.
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The same thing around here.
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That is it, local min, local max, no absolute max, no absolute min.
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Let us go ahead and give some formal definitions to these concepts, because you are often going to see the formal definitions.
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Mathematics is about symbolism.
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We have talked about these things informally, geometrically, let us give them some algebraic identity.
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Formal definitions, we will let f(x) be a function.
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We will let d be its domain.
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Let us define what we mean by absolute max.
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The absolute max also called the global max.
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If there is a number c that is in the domain such that
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the value of f at c is actually bigger than or equal to the value of f(x) for every single x in the domain.
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Then, f achieves its absolute max at c.
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The value f(c) is the absolute maximum value.
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Once again, if there is a number c that has to be in the domain,
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such that f(c) is bigger than f(x) for all the other x in the domain, then f achieves its absolute max at c and f(c) is the absolute max.
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That is it, it is the largest y value that the function takes in the domain.
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This just happens to be the formal definition.
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Let us give a definition for absolute min which is also called a global min.
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You can imagine, it is going to be exact same thing except this inequality is going to be reversed.
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Global min, if there is … everything else is the same.
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Such that floats that f(c) is actually less than or equal to f(x), for all x in d.
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Then, f achieves its absolute min at c and f(c) is that absolute min.
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Nothing strange, completely intuitive.
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You know what is going.
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But again, it is very important.
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To start the formal definitions in mathematics, very important because you want to be very precise
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about what is it that we are talking about.
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We want to be able to take intuitive notions and put them into some symbolic form.
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Let us go ahead and define what we mean by local max and local min.
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You know what, I think I’m going to go all these wonderful colors to choose from.
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I’m going to go to black for this one.
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Local max also called the relative max.
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The definition is, if there is a c in the domain such that f(c) is greater than or equal to f(x),
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for all x not in d, for all x in some open interval around c.
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Remember what we said, once you have a point c, we have to take some interval around this point c.
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If I go to the right of c and to the left of c, that the function drops, that is that.
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Now it is not over everything, it is just locally speaking, a little bit.
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If there is a c and d such that f and c is greater than or equal f(x) for all x in some open interval around c,
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then f achieves a local max at c and f(c) is that local max.
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The definition for local min which is also called a relative min.
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Everything is exactly the same.
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I’m just going to do if … the defining condition is f(c) is less than or equal to f(x) for all x in some open interval around c.
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Then, f achieves its local min at c and f(c) is that local min.
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Nothing strange, let me go back to blue.
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The absolute max value and the absolute min values are also called the extreme values.
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We are going to list a important theorem called the extreme value theorem.
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We have the extreme value theorem.
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If f is continuous on a closed interval ab,
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then f achieves both an absolute max and an absolute min on ab.
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Very important, the two hypotheses of this theorem that, if f is continuous,
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f has to be continuous and it has to be a closed interval.
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If those two hypotheses are satisfied, then the conclusion is that f has an absolute max and an absolute min on that closed interval.
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It might be inside the interval, in other words it might be a local max or local min were achieved its highest.
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Or it might actually be at the endpoints because the closed interval, the endpoint are part of the domain.
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If it is continuous, if it is closed interval, then both absolute max and absolute min are achieved.
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If one or both of the hypotheses are not satisfied, you cannot conclude that it has a max or a min.
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May or may not, but you cannot conclude it.
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If a function is not continuous on its domain, if the interval is not closed, all bets are off.
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Let us go ahead and take a look at a couple of examples of that.
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I will do this in red.
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We have got something like that.
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Let us go something like that.
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Here is a and here is d, this is a closed interval.
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The function is continuous.
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Therefore, it achieves its absolute max and absolute min somewhere on this interval, based on the extreme value theorem.
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In this particular case, here is your absolute min,
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I’m reversing everything today.
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This is your absolute max and this is your absolute min.
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In this particular case for this function, they also happen to be local max and local min but that does not matter.
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Continuous function, closed interval, it achieves an absolute max and it achieves an absolute min.
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Let us look at another graph.
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This is a, this is b, continuous function, closed interval.
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We have the absolute min, we have the absolute max, on that closed interval always.
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Let me draw a little circle, something like that.
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This is a, this is b, this is a closed interval.
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However, the function is not continuous.
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Therefore, it does not achieve both an absolute max and an absolute min.
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Here I see some absolute max but there is no absolute min.
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Because in this particular case, this is an open circle.
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It gets smaller and smaller but we do not know how small it actually gets.
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If there is no absolute smallest value of y on this.
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There is not because it is discontinuous there.
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It does not satisfy the hypotheses, so it does not apply.
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We will do one more.
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We will take the function f(x) = 1/x, your standard hyperbola.
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Here there is no max, there is no absolute max, and there is no absolute min.
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The reason is it is continuous but there is no closed interval.
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I have not specified a closed interval that has well defined endpoints.
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This is going to keep climbing and climbing.
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This is going to keep dropping and dropping.
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You might think yourself, wait a minute, in this particular case, cannot I just say that 0 is an absolute min?
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No, what 0 is a lower bound.
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In other words, the function will never drop below 0.
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But I cannot say that there is a smallest number that is still bigger than 0, that this function will hit.
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It is going to keep getting smaller and smaller, heading towards 0.
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In this case, like that one, this number is a lower bound on this function.
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In other words, it will never be lower than that.
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But that does not mean that that is in absolute minimum because it does not achieve its minimum.
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Because for every number I find that is small, that is close to this lower bound, like close to 0 over here,
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I can find another number smaller than that closer to 0.
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That is the whole idea of this infinite process.
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There is a very big difference.
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A lower bound or upper bound is not the same as absolute max and absolute min.
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Absolute max and absolute min, they have to belong to the domain.
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The x values at which the absolute max and absolute min are achieved, they have to be part of the domain.
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Let us move on, one more theorem here.
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Let us go ahead and leave it in red.
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If f(x) has a local max or a local min at c in the domain of the function, then, f’ at c is equal to 0.
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All that means is the following.
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We already know what local max and local min look like.
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Local min is a valley, local max is a crest.
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If I have some function like this, this is a local max and this is a local min.
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We will call this c1, we will call this c2, whatever the x value happens to be.
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This says that at the local max and at the local min, the slope is 0.
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The derivative f’ at c is 0, the derivative is 0, the slope is 0.
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We can see it geometrically.
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We are going to have a positive slope, from your perspective, if you are moving from negative to positive.
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Positive slope, it is going to hit 0 and it is going to go down like this.
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That tells us that we have a crest, a local max.
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Then this one, local min.
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That is it, that is all this theorem says.
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Let us go ahead and give the definition.
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Definition, something called a critical number or a critical value.
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The number c that is in the domain such that, f’ at = 0 or f’ at c does not exist.
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A critical number or a critical value of the function, it is a number in the domain such that it is a number c in the domain,
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such that f’ at that number is either equal to 0 or f’ at c does not exist.
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In this particular case, these values, f’ of these values is definitely 0, it is a horizontal slope.
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These are critical values.
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An example of one where it does not exist is the absolute value function.
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Absolute value function goes that way and it goes that way.
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It is continuous there at 0 but is not differentiable there.
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Because it is not differentiable there, that is a critical value of the absolute value function.
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F’ at c = 0 or f’ at c does not exist.
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If it is not defined there, that is not considered a critical value.
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It has to actually be defined there.
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It has to be in the domain, that is important.
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If it is not part of the domain, then all bets are off.
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Let me write this a little bit better.
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Let us do it in red.
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The procedure for finding the critical values, very simple.
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The procedure for finding the critical values of a function f(x).
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Find the derivative f’(x) into = 0.
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Set f’(x) equal to 0 and solve for all values of x that satisfy this equation.
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This equation, the only other thing that you have to watch out for is place on the domain
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with the function is not differentiable.
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Other than that, find the derivative, set the derivative equal to 0, and you are done.
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Let us go ahead and actually do an example of this.
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I’m going to call this example 1.
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Example 1, find the critical values of f(x) = x + sin(x).
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We know that f(x) = x + sin(x).
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F’(x) is equal to 1 + cos(x).
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We take the derivative and we set it equal to 0.
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1 + cos(x) is equal to 0 and we solve.
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Cos(x) = -1.
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Therefore, x = π.
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Let us just stick to a particular domain, let us go from 0 to 2π.
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We know that it repeats over and over again but that is fine.
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We will stick to 0 and 2 π.
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In this particular case, in this domain, the critical values are x = π.
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That is a place where the derivative is equal to 0.
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Are there any places where this function is actually not differentiable on this domain?
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No, the cosign function is discontinuous everywhere and it is differentiable everywhere.
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I do not have to worry about that other part of the definition of critical value.
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I just have to worry about taking the derivative and setting it equal to 0, and solving.
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Let us list the procedure for finding.
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Do not worry about it, as far as this example is concerned,
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this particular lesson is just the presentation of the material with a quick example.
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The following lesson is going to be many examples of what it is that we are doing.
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There are going to be plenty examples, I promise you.
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Now we are going to talk about the procedure for finding the absolute max and absolute min.
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Remember, we have that extreme value theorem.
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We said that, if a function is continuous on a closed interval, that it achieves its max and min, absolute max and min.
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How do we find that absolute max and min?
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Here is how you do it.
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Procedure for finding the absolute max and absolute min of f(x) on ab, the closed interval.
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One, find the critical values of f(x), that is what we just did, that procedure.
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Evaluate the original function, evaluate f(x) at each critical value.
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Third step, we want to evaluate f(x), the original function, that is going to be the hardest part,
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especially now that we get into this max and min.
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We are going to be talking more about derivative, setting them equal to 0, using it to graph.
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You are going to be dealing with functions, their derivatives, the first derivative and the second derivative.
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You are going to find certain values and you are going to be plugging them back in.
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Which do you plug it back in?
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You do have to be careful.
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It is an easy procedure but just make sure the values that you find, you are plugging back into the right function.
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When we find the critical values, we are going to be using f’(x), setting it equal to 0.
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When we find those values, we are going to be actually to be putting them back into f(x) not f’(x).
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The third part is evaluate f(x) at each end point.
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In other words, you want to evaluate f(a) and f(b).
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Now you have a list of values.
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You have a list of values f(x) at each critical value.
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You have f(a) and you have f(b).
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Of these tabulated values for f(x), the greatest one, the greatest is the absolute max and the smallest is the absolute min.
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That is it, nice and simple.
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Let us do an example.
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Let me skip this graph.
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Now example 2, what are the absolutely max and absolute min of f(x) is equal to x + 4 sin x on 0 to 2 π.
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Let us go ahead and take f’.
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F’(x) is equal to 1 + 4 × cos(x).
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We set that equal to 0 because our first step is to find the critical values of this function.
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Critical values, we take the derivative and we set it equal to 0.
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I have got cos(x) is equal to 1/4 that means that x is equal to the inverse cos of 1/4 or 0.25.
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On the interval, from 0 to 2 π, I find that x is equal to 1.82 rad or 104.5°, if you prefer degrees.
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Or we have 4.46 rad, I’m sorry this is going to be -1/4, 255.5°.
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These are our critical values.
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We want to evaluate f at those critical values, that is our second step.
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F(1.82) is equal to 5.70.
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In other words, I put these back into the original function to evaluate it.
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F(4.46) is equal to 0.59.
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I’m going to evaluate at the endpoints, 0 and 2 π.
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F(0) is equal to 0 and f(2 π) = 2 π which = 6.28.
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I have 5.7 and 0.59, 0 and 6.28.
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The absolute max is the biggest number among these.
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The absolute max is equal to 6.28.
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The absolute max happens at x = 2 π.
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The absolute min value, that is going to be the smallest number here is 0, and that happens at x = 0.
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That is all, find the critical values, evaluate the function of the critical values.
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Evaluate the function at the endpoints, pick the biggest and pick the smallest.
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You are done, let us see what this looks like.
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Here is the function.
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This right here is your f(x) and I decided to go ahead and draw the derivative on there too.
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This is f’.
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This f(x) here, this is the x + 4 sin x.
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This f’(x), this is equal to 1 + 4 × cos(x).
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You know it achieves its maximum value at whatever it happened to be which was 2 π, I think, and its minimum at 0.
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There you go, that is it.
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Over the domain, you are done.
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2 π would put you on 6.28, somewhere around here.
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Sure enough, that is a the highest value because we are looking at that.
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This is going to be the lowest value, that is all.
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We have the critical values where the derivative equal 0.
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Those were here and here.
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In other words, whatever that was, the 1.82 and I think the 4.46, those are local max and min.
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Local max and local min but they are not absolute max and absolute min.
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We have to include the endpoint.
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In this particular case, it is the end points at which this function achieves its absolute values, its extreme values.
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Thank you so much for joining us here at www.educator.com.
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We will see you next time so that we can do some example problems with maximum and minimum values.
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Take care, bye.