Explanation Given: Function f ( x ) = ( x 3 − 8 ) 4 Formula Used: Critical points of the function are the points where the function is defined and its derivative is undefined or zero. Let us assume that p is the critical point of a continuous function f . Moving from left to right If p′ changes from negative to positive at p , then f has local minimum at p If p′ changes from positive to negative at p , then f has local maximum at p Calculation: Function is given as f ( x ) = ( x 3 − 8 ) 4 Derivative of above function is calculated as follows: f ′ ( x ) = d d x ( x 3 − 8 ) 4 Using chain rule of derivative Let u = x 3 − 8 and f = u 4 f ′ ( x ) = d f d u . d u d x f ′ ( x ) = d d u ( u 4 ) . d d x ( x 3 − 8 ) f ′ ( x ) = ( 4 u 3 ) . ( 3 x 2 − 0 ) f ′ ( x ) = 12 x 2 u 3 f ′ ( x ) = 12 x 2 ( x 3 − 8 ) 3 In order to find the critical points, put f ′ ( x ) = 0 Thus, we have 12 x 2 ( x 3 − 8 ) 3 = 0 12 x 2 = 0 or

Applied Calculus, 6e WileyPLUS + Loose-leaf

6th Edition

ISBN: 9781119408901

Author: Deborah Hughes-Hallett, Andrew M. Gleason, Patti Frazer Lock, Daniel E. Flath

Publisher: Wiley (WileyPLUS Products)

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Chapter 4.1, Problem 24P

To determine

To find all critical points for below function and then use first-derivative test to determine local maxima and minima. Also, check your answer by graphing

f(x)=(x3−8)4

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Chapter 4.1, Problem 23P

Chapter 4.1, Problem 25P

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Chapter 4 Solutions

Applied Calculus, 6e WileyPLUS + Loose-leaf

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To find all critical points for below function and then use first-derivative test to determine local maxima and minima . Also, check your answer by graphing f ( x ) = ( x 3 − 8 ) 4

Solution Summary: The author explains how to find all critical points for a function and then use first-derivative test to determine local maxima and minima.

Concept explainers

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To find all critical points for below function and then use first-derivative test to determine local maxima and minima. Also, check your answer by graphing

f(x)=(x3−8)4

Want to see the full answer?

Chapter 4 Solutions

To find all critical points for below function and then use first-derivative test to determine local maxima and minima . Also, check your answer by graphing f ( x ) = ( x 3 − 8 ) 4

Solution Summary: The author explains how to find all critical points for a function and then use first-derivative test to determine local maxima and minima.

Concept explainers

Videos

To find all critical points for below function and then use first-derivative test to determine local maxima and minima. Also, check your answer by graphing f(x)=(x3−8)4

Want to see the full answer?

Chapter 4 Solutions

To find all critical points for below function and then use first-derivative test to determine local maxima and minima. Also, check your answer by graphing

f(x)=(x3−8)4