   Chapter 4.1, Problem 76E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Show that 5 is a critical number of the function g ( x ) = 2 + ( x − 5 ) 3 but g does not have a local extreme value at 5.

To determine

To show: The critical number of the function g(x)=2+(x5)3 is 5; but it does not have a local extreme value at 5 .

Explanation

Definition used:

A critical number of a function f is a number c, if it satisfies either of the below conditions:

(1) f(c)=0

(2) f(c) does not exist.

Proof:

Find the first derivative of g(x) .

g(x)=ddx(2+(x5)3)

g(x)=3(x5)2 (1)

Substitute x=5 in equation (1),

g(5)=3(55)2g(5)=0

Hence, it satisfies the condition of the definition of critical number

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