Explanation Given Information: The provided function is f ( x ) = x 4 − 240 x 3 + 16 , 200 x 2 − 60 , 000 . Formula Used: The simple power rule to derivative. d d x ( x n ) = n x n − 1 The steps to calculate the critical values of a function Step 1: First find the derivative of the function. Step 2: Equate the derivate of the function to 0 and calculate the possible critical points. Step 3: Put the critical point in the provided function to get the critical values. Calculation: Consider the provided function f ( x ) = x 4 − 240 x 3 + 16 , 200 x 2 − 60 , 000 , Use the simple power rule to differentiate, d y d x = 4 x 3 − 720 x 2 − 32400 x = 4 x ( x 2 − 180 x + 8100 ) = 4 x ( x − 90 ) 2 Equate the above derivative to 0, y ′ = 0 4 x ( x − 90 ) ( x − 90 ) = 0 Evaluate the values of x from the equation: 4 x ( x − 90 ) 2 = 0 x = 0 , 90 Hence, the value(s) of x are x = 0 , 90 . Evaluate the values of the original functions with the critical values: Substitute 0 for x in f ( x ) = x 4 − 240 x 3 + 16 , 200 x 2 − 60 , 000 . y = ( 0 ) 4 − 240 ( 0 ) 3 + 16200 ( 0 ) 2 − 60 , 000 = 0 − 0 + 0 − 60 , 000 = − 60 , 000 Hence, ( 0 , − 60000 ) is a critical point. Substitute 90 for x in f ( x ) = x 4 − 240 x 3 + 16 , 200 x 2 − 60 , 000 . y = ( 90 ) 4 − 240 ( 90 ) 3 + 16200 ( 90 ) 2 − 60 , 000 = 65610000 − 174960000 + 131220000 − 60000 = 21 , 810 , 000 Hence, ( 90 , 21 , 810 , 000 ) is a critical point

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 10.1, Problem 40E

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To calculate: The location of critical points from the derivation of the function f(x)=x4−240x3+16,200x2−60,000 and determine a viewing window that shows all features of the graph of f(x)=x4−240x3+16,200x2−60,000 and sketch the complete graph.

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Chapter 10.1, Problem 39E

Chapter 10.1, Problem 41E

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Chapter 10 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 10.1 - 1. The x- values of critical points are found...Ch. 10.1 - Prob. 2CP Ch. 10.1 - Prob. 3CP Ch. 10.1 - Prob. 4CP Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - 3. Use the graph of in Problem 1 to identify at...Ch. 10.1 - 4. Use the graph of in Problem 2 to identify at...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...

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The location of critical points from the derivation of the function f ( x ) = x 4 − 240 x 3 + 16 , 200 x 2 − 60 , 000 and determine a viewing window that shows all features of the graph of f ( x ) = x 4 − 240 x 3 + 16 , 200 x 2 − 60 , 000 and sketch the complete graph.

Solution Summary: The author explains the steps to calculate the critical values of a function.

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To calculate: The location of critical points from the derivation of the function f(x)=x4−240x3+16,200x2−60,000 and determine a viewing window that shows all features of the graph of f(x)=x4−240x3+16,200x2−60,000 and sketch the complete graph.

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Chapter 10 Solutions