Explanation Given Information: The provided function is f ( x ) = x 3 − 15 x 2 − 16 , 800 x + 80 , 000 . Formula Used: 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. This, in an interval between two critical values, the sign of the derivative at any value in the interval will be the sign of the derivative at all values in the interval. As per the First Derivative Test, The first derivative of the function is evaluated. The first derivative is made equal to zero in order to get the critical points. The values of the critical values are kept inside the original function which gives the critical points. The intervals of the values of x are then evaluated for the relative maximum and minimum. The viewing window is the interval that can be chosen from the critical points. It can differ for every person. 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 equation f ( x ) = x 3 − 15 x 2 − 16 , 800 x + 80 , 000 , Use the simple power rule to differentiate, d y d x = 3 x 2 − 30 x − 16800 = 3 ( x 2 − 10 x + 5600 ) = 3 ( x + 70 ) ( x − 80 ) Equate the above derivative to 0, y ′ = d y d x = 3 ( x + 70 ) ( x − 80 ) = 0 Evaluate the values of x from the equation: 3 ( x + 70 ) ( x − 80 ) = 0 x = 80 , − 70 Hence, the value(s) of x are x = 80 , − 70 . Evaluate the values of the original functions with the critical values: Substitute 80 for x in the function f ( x ) = x 3 − 15 x 2 − 16 , 800 x + 80 , 000 . y = ( 80 ) 3 − 15 ( 80 ) 2 − 16 , 800 ( 80 ) + 80 , 000 = 512000 − 96000 − 1344000 + 80000 = − 848000 Hence, ( 80 , − 848000 ) is a critical point. Substitute − 70 for x in the function f ( x ) = x 3 − 15 x 2 − 16 , 800 x + 80 , 000

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 38E

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To calculate: The location of critical points and determine a viewing window that shows by features of the graph f(x)=x3−15x2−16,800x+80,000.

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Chapter 10.1, Problem 37E

Chapter 10.1, Problem 39E

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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 and determine a viewing window that shows by features of the graph f ( x ) = x 3 − 15 x 2 − 16 , 800 x + 80 , 000 .

Solution Summary: The author calculates the location of critical points and determines a viewing window that shows by features of the graph.

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To calculate: The location of critical points and determine a viewing window that shows by features of the graph f(x)=x3−15x2−16,800x+80,000.

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Chapter 10 Solutions