INSTRUCTORS Raffi Hovasapian John Zhu

Start learning today, and be successful in your academic & professional career. Start Today!

• ## Related Books

 0 answersPost by ROSE LANDRY on March 12, 2015Your writing are always too small to be clearly seen 1 answerLast reply by: Johnny ZamoraFri Jan 10, 2014 1:33 AMPost by Johnny Zamora on January 10, 2014THe interval for example 2 should have been, [-.5 to + infinity] 1 answerLast reply by: Si Jia WenTue Aug 27, 2013 11:04 PMPost by Steve Denton on October 18, 2012My textbook says f"(x)is neg or pos, not f'(x)--notes on concavity. And so he is always testing f"(x). . . . ??

### Concavity

• Concavity: second derivative test
• Take second order derivative of appropriate function
• Set second order derivative = 0
• Solve for critical point x,or “point of inflection”
• Examine second order derivative behavior around critical point to determine concavity
• Be sure to not confuse second order derivative (concavity) with first order derivative (slope)

### Concavity

Find the points of inflection of f(x) = [1/2] x3 − 3x2 + 7x + 11
• f′(x) = [3/2] x2 − 6x + 7
• f"(x) = 3x − 6
• 3x − 6 = 0
• 3x = 6
• x = 2
There is a point of inflection at x = 2
Find the points of inflection of f(x) = [1/12] x4 − [1/2] x2 + 12
• f′(x) = [1/3] x3 − x
• f"(x) = x2 − 1
• x2 − 1 = 0
• (x + 1)(x − 1) = 0
• x = −1, 1
The points of inflection are x = −1 and x = 1
Determine the concavity of f(x) = sinx on the interval (0, π)
• f′(x) = cosx
• f"(x) = −sinx
• −sinx = 0
• sinx = 0
• x = 0, π
• These two points of inflection are at the end of the interval meaning everything in between must have the same concavity.
• f"([(π)/2]) = −sin[(π)/2] = −1
• f"(x) < 0 on the interval
f(x) = sinx is concave down on the interval (0, π)
Determine the concavity of f(x) = ex
• f′(x) = ex
• f"(x) = ex
• There are no points of inflection, ex> 0
f(x) = ex is concave up.
Determine the concavity of f(x) = 3x2 −7x + 21
• f′(x) = 6x − 7
• f"(x) = 6
• No points of inflection and f"(x) > 0
f(x) = 3x2 − 7x + 21 is concave up.
Determine the concavity of f(x) = x3 − 12x
• f′(x) = 3x2 − 12
• f"(x) = 6x
• 6x = 0
• x = 0
• Inflection point at x = 0
• f"(x) < 0 for x < 0
• f"(x) > 0 for x > 0
f(x) = x3 − 12x is concave down on x < 0 and concave up on x > 0
Find any critical points of f(x) = x3 − 12x and determine whether they represent local minimums or maximums.
• f′(x) = 3x2 − 12
• 3x2 − 12 = 0
• x2 − 4 = 0
• x2 = 4
• x = ±2
• The critical points are x = −2 and x = 2
• From the previous problem we know that the function is concave down at x = −2 and concave up at x = 2. This tells us that x = −2 is a local maximum and x = 2 is a local minimum.
Local maximum at x = −2 and a local minimum at x = 2
Find the critical points and the concavity of f(x) = lnx
• f′(x) = [1/x]
• f′(x) is undefined at x = 0. Critical point at x = 0
• f"(x) = −[1/(x2)]
• No change in concavity, f"(x) never reaches zero.
• f"(x) < 0
f(x) = lnx is concave down and has a critical point at x = 0
Determine the concavity of f(x) = x[1/3]
• f′(x) = [1/3] x−[2/3]
• f"(x) = −[2/9] x−[5/3]
• f"(0) and f′(0) are undefined and f"(x) ≠ 0.
• For x < 0, f"(x) > 0
• For x > 0, f"(x) < 0
f(x) = x[1/3] is concave up when x < 0 and concave down when x > 0
Determine the concavity of f(x) = [1/12] x4 − [1/6] x3 − x2 + 17x + 23
• f′(x) = [1/3] x3 − [1/2] x2 − 2x + 17
• f"(x) = x2 − x − 2
• x2 − x − 2 = 0
• (x − 2)(x + 1) = 0
• x = −1 and x = 2 are the points of inflection
• Test points
• f"(−2) = (−2)2 − (−2) − 2 = 4
• f"(0) = (0)2 − 0 − 2 = −2
• f"(3) = 32 − 3 − 2 = 4
f(x) = [1/12] x4 − [1/6] x3 − x2 + 17x + 23 is concave up when x <−1, concave down when −1 < x < 2, and concave up when x > 2

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Concavity

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 0:00
• Concavity: Second Derivative Test 0:06
• Definition
• Example 1
• Example 2 2:51
• Example 3 4:08
• Example 4 5:52