Question

a. Locate the critical points of the given function. b. Use the First Derivative Test to locate the local maximum and minimum values. c. Identify the absolute minimum and maximum values of the function on the given interval (when they exist). f(x) = 4x - 5 tan ?¹ x; [? 2?3, 2?3] Find any minimum values. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. Local minimum/minima is/are located at x = 1/2. (Type an exact answer. Use a comma to separate answers as needed.) B. There are no local minima. c. Find the absolute minimum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The absolute minimum is -7.408. (Round to three decimal places as needed.) B. There is no absolute minimum. Find the absolute maximum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The absolute maximum is 7.408. (Round to three decimal places as needed.) B. There is no absolute maximum. Question is complete. Tap on the red indicators to see incorrect answers.

          a. Locate the critical points of the given function.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute minimum and maximum values of the function on the given interval (when they exist).

f(x) = 4x - 5 tan ?¹ x; [? 2?3, 2?3]

Find any minimum values. Select the correct choice below and, if necessary, fill in the answer box within your choice.

A. Local minimum/minima is/are located at x = 1/2.
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local minima.

c. Find the absolute minimum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice.

A. The absolute minimum is -7.408.
(Round to three decimal places as needed.)
B. There is no absolute minimum.

Find the absolute maximum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice.

A. The absolute maximum is 7.408.
(Round to three decimal places as needed.)
B. There is no absolute maximum.

Question is complete. Tap on the red indicators to see incorrect answers.

Show more…

a. Locate the critical points of the given function.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute minimum and maximum values of the function on the given interval (when they exist).

f(x) = 4x - 5 tan ?¹ x; [? 2?3, 2?3]

Find any minimum values. Select the correct choice below and, if necessary, fill in the answer box within your choice.

A. Local minimum/minima is/are located at x = 1/2.
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local minima.

c. Find the absolute minimum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice.

A. The absolute minimum is -7.408.
(Round to three decimal places as needed.)
B. There is no absolute minimum.

Find the absolute maximum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice.

A. The absolute maximum is 7.408.
(Round to three decimal places as needed.)
B. There is no absolute maximum.

Question is complete. Tap on the red indicators to see incorrect answers.

Added by Daryl W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Justin Swantek

08/28/2023

Step 1

To locate the critical points, we need to find where the derivative of the function is equal to zero or undefined. So, we take the derivative of f(x) as follows: f'(x) = 4 - 5sec^2(x) Setting f'(x) = 0, we get: 4 - 5sec^2(x) = 0 sec^2(x) = 4/5 Taking the Show more…

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a. Locate the critical points of the given function. b. Use the First Derivative Test to locate the local maximum and minimum values. c. Identify the absolute minimum and maximum values of the function on the given interval (when they exist). f(x) = 4x - 5 tan ⁻¹ x; [− 2√3, 2√3] Find any minimum values. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. Local minimum/minima is/are located at x = 1/2. (Type an exact answer. Use a comma to separate answers as needed.) B. There are no local minima. c. Find the absolute minimum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The absolute minimum is -7.408. (Round to three decimal places as needed.) B. There is no absolute minimum. Find the absolute maximum on the given interval, if it exists. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The absolute maximum is 7.408. (Round to three decimal places as needed.) B. There is no absolute maximum.

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Transcript

00:01 All right, hello, in this question we're given this function f of x equals 4x minus 5 arc tangent of x, and on the interval negative 2 root 3 to 2 root 3, we want to find the absolute max and min, but first we want to go ahead and find our critical points.

00:12 So a is find the critical points of this function.

00:14 In order to do that we need to take a derivative, and that derivative is going to be 4 minus 5 times the derivative of arc tangent of x, which is 1 over 1 plus x squared, and then i want to set that equal to 0 and solve.

00:30 So i can move this term to the right side to make it positive and then cross multiply, so i'll get 4 plus 4x squared equals 5.

00:38 So x is going to be plus or minus the square root of 1 over 4, which is going to be plus or minus one half.

00:47 Both of those fall within our domain of interest, so those will both be good candidates for our critical points.

00:53 We're asking part b to use a first derivative test to identify if these are maxes or mins, so we're going to look at different values of x here.

01:01 I don't need to look at these endpoints because i just want to consider those for absolute max and min, but for as far as the sign goes i can look from negative infinity to negative one half, and then between those two critical points, and then from one half to infinity, and i want to see what the sign of f prime is going to be.

01:23 So this function here, if i plug in a very large negative number, this term is going to get very big, and so this term is essentially going to go to 0.

01:32 So i'm going to have 4 minus 0, it's going to be 4, that's going to be a positive value.

01:36 0 lies within there, so i can just plug in 0, and if i do that i'm going to get 4 minus 5, which is negative 1, so that's going to be negative.

01:42 And then again if i plug in a very large number, this term goes to 0, and so i'll get 4, which is a positive number...

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