For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

### Maximum & Minimum Values of a Function

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Maximum & Minimum Values of a Function, Part 1
- Maximum & Minimum Values of a Function, Part 2
- Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
- Function with Local Max & Min but No Absolute Max & Min
- Formal Definitions
- Absolute Maximum
- Absolute Minimum
- Local Maximum
- Local Minimum
- Extreme Value Theorem
- Theorem: f'(c) = 0
- Critical Number (Critical Value)
- Procedure for Finding the Critical Values of f(x)
- Example I: Find the Critical Values of f(x) x + sinx
- Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]

- Intro 0:00
- Maximum & Minimum Values of a Function, Part 1 0:23
- Absolute Maximum
- Absolute Minimum
- Local Maximum
- Local Minimum
- Maximum & Minimum Values of a Function, Part 2 6:11
- Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
- Function with Local Max & Min but No Absolute Max & Min
- Formal Definitions 10:43
- Absolute Maximum
- Absolute Minimum
- Local Maximum
- Local Minimum
- Extreme Value Theorem
- Theorem: f'(c) = 0
- Critical Number (Critical Value)
- Procedure for Finding the Critical Values of f(x)
- Example I: Find the Critical Values of f(x) x + sinx 29:51
- Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π] 35:31

### AP Calculus AB Online Prep Course

### Transcription: Maximum & Minimum Values of a Function

*Hello, and welcome back to www.educator.com, welcome back to AP Calculus.*0000

*Today, we are going to talk about the maximum and minimum values of a function.*0005

*A function on a given domain, it is going to achieve different types of max and min.*0009

*We are going to talk about an absolute max, an absolute min, local max, and local min.*0015

*Let us jump right on in.*0020

*Let us start off by just looking at this graph.*0024

*Let me go ahead and tell you what this graph is.*0026

*It is what the function is actually, that this graph represents.*0030

*Let us go to black here.*0041

*Let us call it f(x).*0043

*Here f(x) is equal to 3x⁴ - 14x³ + 15x².*0044

*The domain is restricted on this one.*0056

*The domain happens to be -0.5.*0061

*X runs from -0.5 and it is less than or equal to 3.5.*0065

*We know that a function is complete, when you actually specify its domain.*0073

*Generally, for the most part, we do not talk about domains but domains really are important.*0078

*In this case, even though this function is defined over the entire real line, we are going to restrict its domain here.*0082

*Notice that this is less than or equal to.*0090

*The endpoints do matter, so it is defined.*0092

*We have a point there and a point there.*0096

*Like I said, we are going to be talking about an absolute max, an absolute min, a local max, and local min.*0100

*I’m going to go through a few of these graphs and talk about it informally.*0106

*And then, I will go ahead and give formal definitions for what they are.*0110

*The formal definitions are there just for you, they are in your book.*0113

*Our discussion right now in the first couple of minutes is going to make clear exactly what these things are.*0117

*The absolute max of a function on a given domain is exactly what it sounds like,*0123

*it is the highest value that f(x) takes on that domain.*0126

*In this particular one, this point right here, which happens to be 3.5 and 33.69.*0131

*This is the absolute max.*0141

*On the domain, it is the highest value that f(x) actually achieves.*0147

*We know that if did not restrict the domain, it would go up into infinity.*0151

*In that case, there is no absolute max, there is no upper limit that we can say.*0157

*But here, we can because we have restricted the domain.*0161

*Let us talk about the absolute minimum.*0165

*The absolute minimum, the lowest value that a function actually takes on its domain happens to be over here.*0168

*This point is 2.5 and -7.81.*0174

*This is an absolute min, it is the lowest value that it takes on its domain.*0181

*Let us talk about something called a local max and local min.*0189

*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.*0192

*Such that, if you take some little interval around that particular x, that f(x) is bigger than every other number.*0202

*Or the f(x) is smaller than every other number.*0213

*In this case, this right here, this point which happens to be the point 1,4, this is a local max.*0216

*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,*0228

*notice the function is lower than this, the function is lower than this.*0235

*For a nice small little region around the point, if at that point the function achieves the maximum value that it can,*0240

*in that little bit, it is a maximum locally speaking.*0248

*Locally meaning some little neighborhood around that point.*0252

*Here this is an absolute max, it is also called the global max.*0255

*Absolute min also called the global min.*0259

*Overall, what is the biggest?*0261

*Locally, that is this.*0263

*This is a local max.*0265

*Interestingly enough, this point right here also happens to be a local min.*0267

*This point 2.5, if I take a little neighborhood around 2.5,*0273

*within that little neighborhood locally around the 2.5, this is the lowest point.*0279

*Because if I move to the right, the function is bigger.*0284

*If I move to the left, the y value of the function is bigger.*0288

*This also happens to be a local min.*0291

*That can happen, a local min can be an absolute min.*0294

*A local max can be an absolute max.*0298

*You can have more than one local min and local max.*0301

*You can have only one absolute max and absolute min.*0305

*You might have no absolute max, no absolute min, but you have a bunch of local max and min.*0308

*You might have no local max and min but you might have an absolute max and absolute min.*0314

*These all kind of combinations.*0318

*In this particular case, we have an absolute maximum that it achieves.*0321

*We have an absolute minimum, also happens to be local minimum.*0327

*We have a local maximum.*0332

*This end point over here, it does not really matter.*0333

*It is the y value is someplace in between.*0336

*That is it, that is what is going on with absolute max, absolutely min, local max, and local min.*0338

*We also speak about the point at which the function achieves its absolute max, local max.*0346

*In this case, this function achieves an absolute max at x = 3.5.*0352

*It achieves an absolute min at a local min at x = 2.5.*0357

*It achieves a local max at 1.*0361

*The values themselves are the y values, 33.694 and -7.81.*0364

*Let us look at another function here.*0372

*Once again, a graph can have all or none of these things.*0379

*Let me go ahead and write that.*0381

*I think I will use blue.*0384

*A graph can have all, some, or none of absolute max, absolute min, and the local max and min.*0390

*In this particular case, let me see, what function I have got here, it looks like the x² function.*0415

*Here we are looking at the function y = x².*0422

*Once again, we have restricted its domain.*0426

*2x being greater than or equal to 0.*0429

*This point is absolutely included and this just goes off to positive infinity.*0433

*In this particular case, is there a highest point on this domain?*0439

*No, because this goes up into infinity, we cannot say that there is an absolute max.*0444

*There is no absolute max.*0449

*Is there an absolute min?*0454

*Yes, there is, this is the lowest point overall on this domain.*0455

*It is lowest y value.*0460

*This point 0,0 is the absolute min.*0462

*Are there local max or min?*0467

*No, there are not.*0470

*There is no local max, there is no local min.*0472

*You might think yourself, could this not be considered a local minimum?*0478

*No, a local minimum requires that at a point, in this particular case 0,0,*0482

*that there is actually an interval to the left and to the right of it, which satisfies the conditions.*0487

*In other words, yes, if I move to the right, it is true.*0494

*The function gets higher in value but it is not defined to the left.*0497

*It is not defined to the left but for a local, I need something that is defined around.*0501

*An interval around it has to surround that thing.*0507

*The local min always looks like a little valley.*0511

*A local max always looks like a crest of the hill.*0514

*That is it, that is local max and local min.*0518

*In this case, absolute min and no absolute max, no local max, no local min.*0522

*Let us take a look at another function here.*0528

*This particular function right here, we have not restricted the domain at all.*0541

*This goes off to positive infinity, this goes down to negative infinity.*0545

*It looks like some sort of a cubic function.*0549

*I actually did not write down what function this is but that is not a problem.*0552

*We are here to identify graphically absolute max and min, and local max and min.*0555

*In this particular case, there is no absolute max and there is no absolute min.*0560

*However, we do have a local min and we do have a local max.*0572

*Yes, there is a local minimum and it looks like it achieves that minimum at x = 2.*0577

*There is a local maximum and it looks like it achieves that maximum at x = -2.*0583

*The maximum values and the minimum values happen to be the y values.*0591

*Whatever that happens to be, it looks like somewhere around 16, something like that.*0596

*The same thing around here.*0601

*That is it, local min, local max, no absolute max, no absolute min.*0603

*Let us go ahead and give some formal definitions to these concepts, because you are often going to see the formal definitions.*0616

*Mathematics is about symbolism.*0621

*We have talked about these things informally, geometrically, let us give them some algebraic identity.*0623

*Formal definitions, we will let f(x) be a function.*0650

*We will let d be its domain.*0672

*Let us define what we mean by absolute max.*0680

*The absolute max also called the global max.*0685

*If there is a number c that is in the domain such that*0692

*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.*0710

*Then, f achieves its absolute max at c.*0726

*The value f(c) is the absolute maximum value.*0741

*Once again, if there is a number c that has to be in the domain,*0753

*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.*0757

*That is it, it is the largest y value that the function takes in the domain.*0769

*This just happens to be the formal definition.*0774

*Let us give a definition for absolute min which is also called a global min.*0778

*You can imagine, it is going to be exact same thing except this inequality is going to be reversed.*0784

*Global min, if there is … everything else is the same.*0791

*Such that floats that f(c) is actually less than or equal to f(x), for all x in d.*0802

*Then, f achieves its absolute min at c and f(c) is that absolute min.*0824

*Nothing strange, completely intuitive.*0847

*You know what is going.*0849

*But again, it is very important.*0850

*To start the formal definitions in mathematics, very important because you want to be very precise*0853

*about what is it that we are talking about.*0860

*We want to be able to take intuitive notions and put them into some symbolic form.*0861

*Let us go ahead and define what we mean by local max and local min.*0868

*You know what, I think I’m going to go all these wonderful colors to choose from.*0872

*I’m going to go to black for this one.*0877

*Local max also called the relative max.*0883

*The definition is, if there is a c in the domain such that f(c) is greater than or equal to f(x),*0893

*for all x not in d, for all x in some open interval around c.*0914

*Remember what we said, once you have a point c, we have to take some interval around this point c.*0927

*If I go to the right of c and to the left of c, that the function drops, that is that.*0949

*Now it is not over everything, it is just locally speaking, a little bit.*0955

*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,*0960

*then f achieves a local max at c and f(c) is that local max.*0970

*The definition for local min which is also called a relative min.*0989

*Everything is exactly the same.*0998

*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.*0999

*Then, f achieves its local min at c and f(c) is that local min.*1013

*Nothing strange, let me go back to blue.*1038

*The absolute max value and the absolute min values are also called the extreme values.*1045

*We are going to list a important theorem called the extreme value theorem.*1075

*We have the extreme value theorem.*1090

*If f is continuous on a closed interval ab,*1106

*then f achieves both an absolute max and an absolute min on ab.*1126

*Very important, the two hypotheses of this theorem that, if f is continuous,*1152

*f has to be continuous and it has to be a closed interval.*1158

*If those two hypotheses are satisfied, then the conclusion is that f has an absolute max and an absolute min on that closed interval.*1163

*It might be inside the interval, in other words it might be a local max or local min were achieved its highest.*1176

*Or it might actually be at the endpoints because the closed interval, the endpoint are part of the domain.*1182

*If it is continuous, if it is closed interval, then both absolute max and absolute min are achieved.*1188

*If one or both of the hypotheses are not satisfied, you cannot conclude that it has a max or a min.*1195

*May or may not, but you cannot conclude it.*1201

*If a function is not continuous on its domain, if the interval is not closed, all bets are off.*1204

*Let us go ahead and take a look at a couple of examples of that.*1211

*I will do this in red.*1214

*We have got something like that.*1216

*Let us go something like that.*1221

*Here is a and here is d, this is a closed interval.*1230

*The function is continuous.*1234

*Therefore, it achieves its absolute max and absolute min somewhere on this interval, based on the extreme value theorem.*1236

*In this particular case, here is your absolute min,*1245

*I’m reversing everything today.*1249

*This is your absolute max and this is your absolute min.*1251

*In this particular case for this function, they also happen to be local max and local min but that does not matter.*1259

*Continuous function, closed interval, it achieves an absolute max and it achieves an absolute min.*1265

*Let us look at another graph.*1272

*This is a, this is b, continuous function, closed interval.*1280

*We have the absolute min, we have the absolute max, on that closed interval always.*1286

*Let me draw a little circle, something like that.*1311

*This is a, this is b, this is a closed interval.*1317

*However, the function is not continuous.*1322

*Therefore, it does not achieve both an absolute max and an absolute min.*1325

*Here I see some absolute max but there is no absolute min.*1332

*Because in this particular case, this is an open circle.*1336

*It gets smaller and smaller but we do not know how small it actually gets.*1340

*If there is no absolute smallest value of y on this.*1348

*There is not because it is discontinuous there.*1353

*It does not satisfy the hypotheses, so it does not apply.*1356

*We will do one more.*1366

*We will take the function f(x) = 1/x, your standard hyperbola.*1370

*Here there is no max, there is no absolute max, and there is no absolute min.*1379

*The reason is it is continuous but there is no closed interval.*1385

*I have not specified a closed interval that has well defined endpoints.*1391

*This is going to keep climbing and climbing.*1397

*This is going to keep dropping and dropping.*1399

*You might think yourself, wait a minute, in this particular case, cannot I just say that 0 is an absolute min?*1403

*No, what 0 is a lower bound.*1408

*In other words, the function will never drop below 0.*1414

*But I cannot say that there is a smallest number that is still bigger than 0, that this function will hit.*1418

*It is going to keep getting smaller and smaller, heading towards 0.*1425

*In this case, like that one, this number is a lower bound on this function.*1429

*In other words, it will never be lower than that.*1434

*But that does not mean that that is in absolute minimum because it does not achieve its minimum.*1437

*Because for every number I find that is small, that is close to this lower bound, like close to 0 over here,*1442

*I can find another number smaller than that closer to 0.*1448

*That is the whole idea of this infinite process.*1452

*There is a very big difference.*1455

*A lower bound or upper bound is not the same as absolute max and absolute min.*1457

*Absolute max and absolute min, they have to belong to the domain.*1461

*The x values at which the absolute max and absolute min are achieved, they have to be part of the domain.*1468

*Let us move on, one more theorem here.*1480

*Let us go ahead and leave it in red.*1483

*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.*1488

*All that means is the following.*1522

*We already know what local max and local min look like.*1525

*Local min is a valley, local max is a crest.*1527

*If I have some function like this, this is a local max and this is a local min.*1531

*We will call this c1, we will call this c2, whatever the x value happens to be.*1539

*This says that at the local max and at the local min, the slope is 0.*1544

*The derivative f’ at c is 0, the derivative is 0, the slope is 0.*1551

*We can see it geometrically.*1557

*We are going to have a positive slope, from your perspective, if you are moving from negative to positive.*1558

*Positive slope, it is going to hit 0 and it is going to go down like this.*1563

*That tells us that we have a crest, a local max.*1567

*Then this one, local min.*1571

*That is it, that is all this theorem says.*1573

*Let us go ahead and give the definition.*1577

*Definition, something called a critical number or a critical value.*1583

*The number c that is in the domain such that, f’ at = 0 or f’ at c does not exist.*1605

*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,*1631

*such that f’ at that number is either equal to 0 or f’ at c does not exist.*1636

*In this particular case, these values, f’ of these values is definitely 0, it is a horizontal slope.*1645

*These are critical values.*1652

*An example of one where it does not exist is the absolute value function.*1656

*Absolute value function goes that way and it goes that way.*1660

*It is continuous there at 0 but is not differentiable there.*1667

*Because it is not differentiable there, that is a critical value of the absolute value function.*1675

*F’ at c = 0 or f’ at c does not exist.*1683

*If it is not defined there, that is not considered a critical value.*1687

*It has to actually be defined there.*1690

*It has to be in the domain, that is important.*1692

*If it is not part of the domain, then all bets are off.*1697

*Let me write this a little bit better.*1710

*Let us do it in red.*1713

*The procedure for finding the critical values, very simple.*1718

*The procedure for finding the critical values of a function f(x).*1723

*Find the derivative f’(x) into = 0.*1735

*Set f’(x) equal to 0 and solve for all values of x that satisfy this equation.*1751

*This equation, the only other thing that you have to watch out for is place on the domain*1771

*with the function is not differentiable.*1778

*Other than that, find the derivative, set the derivative equal to 0, and you are done.*1782

*Let us go ahead and actually do an example of this.*1789

*I’m going to call this example 1.*1792

*Example 1, find the critical values of f(x) = x + sin(x).*1797

*We know that f(x) = x + sin(x).*1820

*F’(x) is equal to 1 + cos(x).*1827

*We take the derivative and we set it equal to 0.*1833

*1 + cos(x) is equal to 0 and we solve.*1836

*Cos(x) = -1.*1841

*Therefore, x = π.*1846

*Let us just stick to a particular domain, let us go from 0 to 2π.*1852

*We know that it repeats over and over again but that is fine.*1857

*We will stick to 0 and 2 π.*1860

*In this particular case, in this domain, the critical values are x = π.*1862

*That is a place where the derivative is equal to 0.*1869

*Are there any places where this function is actually not differentiable on this domain?*1872

*No, the cosign function is discontinuous everywhere and it is differentiable everywhere.*1876

*I do not have to worry about that other part of the definition of critical value.*1883

*I just have to worry about taking the derivative and setting it equal to 0, and solving.*1887

*Let us list the procedure for finding.*1896

*Do not worry about it, as far as this example is concerned,*1902

*this particular lesson is just the presentation of the material with a quick example.*1905

*The following lesson is going to be many examples of what it is that we are doing.*1910

*There are going to be plenty examples, I promise you.*1916

*Now we are going to talk about the procedure for finding the absolute max and absolute min.*1919

*Remember, we have that extreme value theorem.*1924

*We said that, if a function is continuous on a closed interval, that it achieves its max and min, absolute max and min.*1925

*How do we find that absolute max and min?*1934

*Here is how you do it.*1935

*Procedure for finding the absolute max and absolute min of f(x) on ab, the closed interval.*1936

*One, find the critical values of f(x), that is what we just did, that procedure.*1955

*Evaluate the original function, evaluate f(x) at each critical value.*1970

*Third step, we want to evaluate f(x), the original function, that is going to be the hardest part,*1988

*especially now that we get into this max and min.*1995

*We are going to be talking more about derivative, setting them equal to 0, using it to graph.*1996

*You are going to be dealing with functions, their derivatives, the first derivative and the second derivative.*2002

*You are going to find certain values and you are going to be plugging them back in.*2007

*Which do you plug it back in?*2010

*You do have to be careful.*2012

*It is an easy procedure but just make sure the values that you find, you are plugging back into the right function.*2013

*When we find the critical values, we are going to be using f’(x), setting it equal to 0.*2020

*When we find those values, we are going to be actually to be putting them back into f(x) not f’(x).*2025

*The third part is evaluate f(x) at each end point.*2032

*In other words, you want to evaluate f(a) and f(b).*2045

*Now you have a list of values.*2057

*You have a list of values f(x) at each critical value.*2061

*You have f(a) and you have f(b).*2064

*Of these tabulated values for f(x), the greatest one, the greatest is the absolute max and the smallest is the absolute min.*2072

*That is it, nice and simple.*2099

*Let us do an example.*2102

*Let me skip this graph.*2130

*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.*2132

*Let us go ahead and take f’.*2171

*F’(x) is equal to 1 + 4 × cos(x).*2173

*We set that equal to 0 because our first step is to find the critical values of this function.*2180

*Critical values, we take the derivative and we set it equal to 0.*2184

*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.*2189

*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.*2194

*Or we have 4.46 rad, I’m sorry this is going to be -1/4, 255.5°.*2218

*These are our critical values.*2233

*We want to evaluate f at those critical values, that is our second step.*2237

*F(1.82) is equal to 5.70.*2242

*In other words, I put these back into the original function to evaluate it.*2249

*F(4.46) is equal to 0.59.*2256

*I’m going to evaluate at the endpoints, 0 and 2 π.*2264

*F(0) is equal to 0 and f(2 π) = 2 π which = 6.28.*2271

*I have 5.7 and 0.59, 0 and 6.28.*2283

*The absolute max is the biggest number among these.*2289

*The absolute max is equal to 6.28.*2294

*The absolute max happens at x = 2 π.*2300

*The absolute min value, that is going to be the smallest number here is 0, and that happens at x = 0.*2309

*That is all, find the critical values, evaluate the function of the critical values.*2322

*Evaluate the function at the endpoints, pick the biggest and pick the smallest.*2327

*You are done, let us see what this looks like.*2330

*Here is the function.*2336

*This right here is your f(x) and I decided to go ahead and draw the derivative on there too.*2339

*This is f’.*2346

*This f(x) here, this is the x + 4 sin x.*2352

*This f’(x), this is equal to 1 + 4 × cos(x).*2359

*You know it achieves its maximum value at whatever it happened to be which was 2 π, I think, and its minimum at 0.*2369

*There you go, that is it.*2377

*Over the domain, you are done.*2378

*2 π would put you on 6.28, somewhere around here.*2383

*Sure enough, that is a the highest value because we are looking at that.*2394

*This is going to be the lowest value, that is all.*2403

*We have the critical values where the derivative equal 0.*2407

*Those were here and here.*2410

*In other words, whatever that was, the 1.82 and I think the 4.46, those are local max and min.*2414

*Local max and local min but they are not absolute max and absolute min.*2420

*We have to include the endpoint.*2424

*In this particular case, it is the end points at which this function achieves its absolute values, its extreme values.*2427

*Thank you so much for joining us here at www.educator.com.*2436

*We will see you next time so that we can do some example problems with maximum and minimum values.*2438

*Take care, bye.*2444

1 answer

Last reply by: Professor Hovasapian

Fri Apr 7, 2017 9:57 PM

Post by Daniel Persaud on February 24 at 11:52:36 PM

For the critical value question how did you get x = pi. should it not be -1

1 answer

Last reply by: Professor Hovasapian

Thu Apr 7, 2016 2:09 AM

Post by Zhe Tian on April 2, 2016

For the first graph, wouldn't there be a local minimum at x=0?

1 answer

Last reply by: Professor Hovasapian

Thu Dec 3, 2015 12:55 AM

Post by Gautham Padmakumar on November 28, 2015

You made a small writing typo error in example 2 you wrote down cos x = 1/4 where its actually cos x = -1/4 but the values for x are right anyways!