# Whether the statement, if f has an absolute minimum value at c then f ′ ( c ) = 0 is true or false.

### 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 4, Problem 2RQ
To determine

## Whether the statement, if f has an absolute minimum value at c then f′(c)=0 is true or false.

Expert Solution

The given statement is false.

### Explanation of Solution

Reason:

The given statement as the below example disproves the given statement.

Consider the function, f(x)=|x| .

As f is always non-negative function it has absolute minimum at x=0 .

But derivative of the function does not exist at x=0

Calculate the derivative of f(x)=|x| as follows.

ddx|x|=limh0|x+h||x|h (1)

limh0+|x+h||x|h=limh0+x+hxh=limh0+hh=1

And similarly,

limh0|x+h||x|h=limh0+xh+xh=limh0+hh=1

Thus the left hand limit is not same as the right hand limit.

So, f'(0) does not exist.

Therefore, the given statement is false.

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