# That the value of f ( 0 ) is equal to 0 and f ' ( 0 ) is equal to 0 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter 2, Problem 17P
To determine

## To find:That the value of f(0) is equal to 0 and f'(0) is equal to 0 .

Expert Solution

The value of f(0) is equal to 0 and f'(0) is equal to 0 is proved.

### Explanation of Solution

Given information:

Suppose f is a function with the property that |f(x)|x2 for all x .

Calculation:

Let x0 , then x0 , so we can write the inequality |f(x)|x2 as.

|f(x)|xx , for all xR

Using ||f(x)0|x||x| or 0|f(x)f(0)x||x| for f(0)=0 .

Apply limits to the inequality.

limx0limx|f(x)f(0)x0|limx|x|0limx|f(x)f(0)x0|0limx|f(x)f(0)x0|=0|limxf(x)f(0)x0|=0

limxf(x)f(0)x0=0f'(0)=0

Therefore, the value of f(0) is equal to 0 and f'(0) is equal to 0 is proved.

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