Explanation Given: Function f ( x ) = x x 2 + 1 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 x 2 + 1 Derivative of above function is calculated as follows: f ′ ( x ) = d d x ( x x 2 + 1 ) Using quotient rule of derivative f ′ ( x ) = d d x ( x ) × ( x 2 + 1 ) − ( x ) × d d x ( x 2 + 1 ) ( x 2 + 1 ) 2 f ′ ( x ) = ( 1 ) × ( x 2 + 1 ) − ( x ) × ( 2 x + 0 ) ( x 2 + 1 ) 2 f ′ ( x ) = x 2 + 1 − 2 x 2 ( x 2 + 1 ) 2 f ′ ( x ) = 1 − x 2 ( x 2 + 1 ) 2 In order to find the critical points, put f ′ ( x ) = 0 Thus, we have 1 − x 2 (

6th Edition

ISBN: 9781119275565

Author: Hughes-Hallett

Publisher: WILEY

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Chapter 4.1, Problem 25P

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)=xx2+1

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Chapter 4.1, Problem 24P

Chapter 4.1, Problem 26P

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Chapter 4 Solutions

APPLIED CALCULUS-PRINT COMPANION (LL)

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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 x 2 + 1

Solution Summary: The author explains that critical points of a function are the points where the function is defined and its derivative is undefined or zero.

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)=xx2+1

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 x 2 + 1

Solution Summary: The author explains that critical points of a function are the points where the function is defined and its derivative is undefined or zero.

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)=xx2+1

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)=xx2+1