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In Problems 31-36, both a function and its derivative are given. Use them to find critical values, critical points, intervals on which the function is increasing and decreasing, relative maxima, relative minima, and horizontal points of inflection; sketch the graph of each function. f ( x ) = x − 3 x 2 / 3 f ' ( x ) = x 1 / 3 − 2 x 1 / 3

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Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781337625340

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BuyFindarrow_forward

Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781337625340
Chapter 10.1, Problem 36E
Textbook Problem
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In Problems 31-36, both a function and its derivative are given. Use them to find critical values, critical points, intervals on which the function is increasing and decreasing, relative maxima, relative minima, and horizontal points of inflection; sketch the graph of each function.

f ( x ) = x 3 x 2 / 3    f ' ( x ) = x 1 / 3 2 x 1 / 3

To determine

To calculate: The critical values, critical points, the relative maxima, relative minima, or horizontal points of inflection of the function y=x3x23 and the derivative dydx=x132x13 and also sketch the graph.

Explanation of Solution

Given Information:

The provided function is y=x3x23 and the derivative is dydx=x132x13.

Formula Used:

To obtain the relative maxima and minima of a function,

1. Find the first derivative of the function.

2. Set the derivative equal to 0 to obtain the critical points.

3. Evaluate f(x) at values of x to the left and one to the right of each critical point to develop a sign diagram.

(a) If f(x)>0 to the left and f(x)<0 to the right of the critical value, the critical point is a relative maximum point.

(b) If f(x)<0 to the left and f(x)>0 to the right of the critical value, the critical point is a relative minimum point.

Calculation:

Consider the provided function y=x3x23,

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

Hence, there will no change in the values of critical values as in the derivative graph.

Consider the provided derivative,

dydx=x132x13

Put the value of y=0,

y=dydx=x132x13=0

Evaluate the values of x from the equation:

x132x13=0x=0,8

Hence, the value(s) of x are x=0,8.

Evaluate the values of the original functions with the critical values:

Put x=0 in the equation y=x3x23,

y=03(0)23=0

Hence, (0,0) is a critical point.

Put x=8 in the equation y=x3x23,

y=83(8)23=83(4)=4

Hence, (8,4) is a critical point.

Test the derivative at the left and right sides of all the critical points

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Chapter 10 Solutions

Mathematical Applications for the Management, Life, and Social Sciences
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Inventory cost model A company needs to...Ch. 10 - 54. Average cost Suppose the total cost of...Ch. 10 - 55. Market share Suppose a company’s percent share...Ch. 10 - Find the relative maxima, relative minima, points...Ch. 10 - Find the relative maxima, relative minima, points...Ch. 10 - Find the relative maxima, relative minima, points...Ch. 10 - In Problems 4-6, use the function . 4. Over what...Ch. 10 - In Problems 4-6, use the function . 5. Find the...Ch. 10 - In Problems 4-6, use the function . 6. Find the...Ch. 10 - 7. Find the absolute maximum and minimum for on...Ch. 10 - 8. Find all horizontal and vertical asymptotes of...Ch. 10 - 9. Use the following graph of y = f(x) and the...Ch. 10 - 10. Use the following figure to complete parts...Ch. 10 - 11. If , and , what can we conclude about the...Ch. 10 - 12. The aged dependency ratio is defined as the...Ch. 10 - 13. The revenue function for a product is dollars...Ch. 10 - 14. The cost of producing x units of a product is...Ch. 10 - 15. A firm sells 100 TVs per month at $300 each,...Ch. 10 - 16. An open-top box is made by cutting squares...Ch. 10 - 17. A company estimates that it will need 784,000...

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