# Whether the statement, if f ′ ( c ) = 0 , then f has a local maximum or minimum at c 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 1RQ
To determine

## Whether the statement, if f′(c)=0 , then f has a local maximum or minimum at c is true or false.

Expert Solution

The given statement is false.

### Explanation of Solution

Reason:

If f(c)=0 then the tangent line of f will be parallel to the x-axis and it does not guarantee of the existence of local maximum or local minimum.

For example, consider the function f(x)=x3 .

Differentiate with respect to x.

f(x)=ddx(x3)=3x2

Set f(x)=0 to obtain,

3x2=0x=0

For all non-zero values of x, f(x) is increasing thus x=c is an inflection point but not local maximum or minimum.

Therefore, the given statement is false.

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