BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4, Problem 26RE

(a)

To determine

To find:

The number of g have an extreme value.

Expert Solution

Answer to Problem 26RE

  g=0

  whenx=0

Explanation of Solution

Given:

The function

  g(x)=f(x2)

  f(x)<0forx<0andf(x)>0forx>0

Concept used:

The left derivative and right derivative of a function f at a point x=c are , equal

  f'(c)=limh0f(ch)f(c)hf+'(c)=limh0+f(c+h)f(c)h

Calculation:

  g(x)=f(x2)..................(1)

Differentiating equation (1) with respect to x

  g(x)=2xf(x2).....................(2)

Differentiating equation (1) with respect to x by the product and chain rules

  g(x)=2xf(x2)g(x)=(2x)f(x2)+2x[f(x2)]g(x)=2f(x2)+2x[2xf(x2)]g(x)=2f(x2)+4x2[f(x2)]

  g=0

  whenx=0

(b)

To determine

To find:

The number of g have the concavity .

Expert Solution

Answer to Problem 26RE

  g is concave up for all real numbers.

Explanation of Solution

Given:

The function

  g(x)=f(x2)

  f(x)<0forx<0andf(x)>0forx>0

Concept used:

The left derivative and right derivative of a function f at a point x=c are , equal

  f'(c)=limh0f(ch)f(c)hf+'(c)=limh0+f(c+h)f(c)h

Calculation:

  g(x)=f(x2)..................(1)

Differentiating equation (1) with respect to x

  g(x)=2xf(x2).....................(2)

Differentiating equation (1) with respect to x by the product and chain rules

  g(x)=2xf(x2)g(x)=(2x)f(x2)+2x[f(x2)]g(x)=2f(x2)+2x[2xf(x2)]g(x)=2f(x2)+4x2[f(x2)]

Given that

  f(x)<0forx<0andf(x)>0forx>0

  x2 is positive for x0,sof(x2)>0forx0

  g=0 ;x=0

Since g is positive on either side that g is concave up for all real numbers.

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