Explanation Given Information: The provided function is f ( x ) = x 3 − 225 x 2 + 15 , 000 x − 12 , 000 . 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 f ( x ) = x 3 − 225 x 2 + 15 , 000 x − 12 , 000 , Use the simple power rule to differentiate, d y d x = 3 x 2 − 450 x + 15000 = 3 ( x 2 − 150 x + 5000 ) = 3 ( x − 100 ) ( x − 50 ) Equate the above derivative to 0, y ′ = d y d x = 3 ( x − 100 ) ( x − 50 ) = 0 Evaluate the values of x from the equation: 3 ( x − 100 ) ( x − 50 ) = 0 x = 100 , 50 Hence, the value(s) of x are x = 100 , 50 . Evaluate the values of the original functions with the critical values: Substitute 100 for x in the function f ( x ) = x 3 − 15 x 2 − 16 , 800 x + 80 , 000 . y = ( 100 ) 3 − 225 ( 100 ) 2 + 15 , 000 ( 100 ) − 12 , 000 = 1000000 − 2250000 + 1500000 − 12000 = 238 , 000 Hence, ( 100 , 238000 ) is a critical point. Substitute 50 for x in the function f ( x ) = x 3 − 15 x 2 − 16 , 800 x + 80 , 000 . y = ( 50 ) 3 − 225 ( 50 ) 2 + 15 , 000 ( 50 ) − 12 , 000 = 125000 − 225 ( 2500 ) + 15000 ( 50 ) − 12000 = 300500 Hence, ( 50 , 300500 ) is a critical point

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 10.1, Problem 37E

To determine

To calculate: The location of critical points and determines a viewing window that shows all features of the graph f(x)=x3−225x2+15,000x−12,000 also sketch the graph.

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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 determines a viewing window that shows all features of the graph f ( x ) = x 3 − 225 x 2 + 15 , 000 x − 12 , 000 also sketch the graph.

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

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To calculate: The location of critical points and determines a viewing window that shows all features of the graph f(x)=x3−225x2+15,000x−12,000 also sketch the graph.

Chapter 10 Solutions