# Whether the statement, if f ′ ′ ( 2 ) = 0 then ( 2 , f ( 2 ) ) is an inflection point of the curve y = f ( x ) 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 6RQ
To determine

## Whether the statement, if f′′(2)=0 then (2,f(2)) is an inflection point of the curve y=f(x) is true or false.

Expert Solution

The given statement is false.

### Explanation of Solution

Reason:

Result used:

Inflection point occurs at x=a only if the following conditions are satisfied.

1) The function f must be continuous at x=a

2) The second derivative f(x) must change its sign.

The given statement is false since the following example disproves the given statement.

Consider a function, y=(x2)4 .

Let f(x)=(x2)4 .

Differentiate f(x) with respect to x.

f(x)=4(x2)3

Find the second derivative of f(x) .

f(x)=12(x2)2

Substitute x=2 in f''(x) .

f(2)=12(22)2=0

Thus, f(x) do not changes its sign.

Hence, (2,f(2)) is not an inflection point.

Therefore, the given statement is false.

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