   Chapter 3.R, Problem 37E

Chapter
Section
Textbook Problem

# Let g ( x ) = f ( x 2 ) , where f is twice differentiable for all x, f ′ ( x ) > 0 for all x ≠ 0 , and f is concave downward on ( − ∞ , 0 ) and concave upward on ( 0 , ∞ ) .(a) At what numbers does g have an extreme value?(b) Discuss the concavity of g.

To determine

Part (a)

To find:

The numbers where the function g has extreme value

Explanation

1) Concept:

Use thefirst derivative test and critical numbers.

First Derivative Test:

Suppose that c is a critical number of a continuous function f defined on an interval.

i. If f'x>0 for all x<c and f'x<0 for all x>c, then fc is the absolute maximum value of f

ii. If f'x<0 for all x<c and f'x>0 for all >c, then fc is the absolute minimum value of f.

Critical number:

A critical number of a function f   is a number c in the domain of f  such that either  f'c=0 or f'c does not exist.

2) Given:

gx=f(x2), f  is twice differential for all x . f'x>0 for x0, f  is concave downward on -,0 and concave upward on 0,

To determine

Part (b):

To discuss:

The concavity of g

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