Explanation Given: The function is: g ( x ) = 2 x e − x / 3 . Formula used: Write the formula for derivative of power rule. If f ( x ) = x n for any number n , then the derivative. f ′ ( x ) = n x n − 1 If f ( x ) = k for any real number k , then the derivative is defined by the formula. f ′ ( x ) = 0 Write the formula for derivative of exponents. f ( x ) = e n x for any number n , then the derivative. f ′ ( x ) = n e n x To find local maximum and minimum find the values of x from the equation f ′ ( x ) = 0 . Then find f ″ ( x ) and put the values of x . If f ″ ( x ) equals some positive quantity then it implies Local minimum is located at point x and if f ″ ( x ) equals some negative quantity then Local maximum is located at that point. Calculation: Consider the given function: g ( x ) = 2 x e − x / 3 …… (1) Differentiate g ( x ) with respect to x . d d x ( g ( x ) ) = d d x ( 2 x e − x / 3 ) g ′ ( x ) = ( 2 x ) d d x ( e − x / 3 ) + ( e − x / 3 ) d d x ( 2 x ) g ′ ( x ) = 2 x e − x / 3 ( − 1 3 ) + ( e − x / 3 ) ( 2 ) g ′ ( x ) = − 2 x e − x / 3 3 + 2 e − x / 3 Solve further as, g ′ ( x ) = 2 e − x / 3 ( − x 3 + 1 ) …… (2) Now, find the values of x for which g ′ ( x ) becomes zero. g ′ ( x ) = 0 2 e − x / 3 ( − x 3 + 1 ) = 0 Solve for x . 2 e − x / 3 = 0 or ( − x 3 + 1 ) = 0 e − x / 3 = 0 or − x 3 = − 1 e − x / 3 = e − ∞ or − x = − 1 ( 3 ) x = ∞ or x = 3

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12.3, Problem 23E

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To calculate: The locations of absolute extrema of the function given below on the given interval:

g(x)=2xe−x/3 ;(−∞,∞)

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Chapter 12.3, Problem 22E

Chapter 12.3, Problem 24E

Chapter 12 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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The locations of absolute extrema of the function given below on the given interval: g ( x ) = 2 x e − x / 3 ; ( − ∞ , ∞ )

Solution Summary: The author explains how the function g(x) has no point of absolute minima and an absolute maximum.

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To calculate: The locations of absolute extrema of the function given below on the given interval:

g(x)=2xe−x/3 ;(−∞,∞)

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Chapter 12 Solutions

The locations of absolute extrema of the function given below on the given interval: g ( x ) = 2 x e − x / 3 ; ( − ∞ , ∞ )

Solution Summary: The author explains how the function g(x) has no point of absolute minima and an absolute maximum.

Concept explainers

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To calculate: The locations of absolute extrema of the function given below on the given interval: g(x)=2xe−x/3 ;(−∞,∞)

Want to see the full answer?

Chapter 12 Solutions

To calculate: The locations of absolute extrema of the function given below on the given interval:

g(x)=2xe−x/3 ;(−∞,∞)