- #1

dozzer

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## Homework Statement

f is a continuous function on [0, 8] and satisfies the following:

Second Derivative Sign Test

**x**;

**f''**

0 [tex]\leq[/tex] x < 3 ; -

3 ; 0

3 < x < 5 ; +

5 ; Does Not Exist

5 < x < 6 ; -

6 ; 0

6 < x [tex]\leq[/tex] 8 ; -

Based on this information is there a point of inflection at x = 5?

(a) Definitely

(b) Possibly

(c) Definitely not

## Homework Equations

Definition: A point of inflection is where the concavity changes and there is a tangent line.

Concavity changes when f'' changes signs. Negative f'' means concave down; positive f'' means concave up.

## The Attempt at a Solution

There is a sign change at f''(5), but the correct answer could either be "definitely" or "possibly." The point of inflection does not exist, but there is still a sign change, meaning it is a point of inflection. But my class and I were wondering whether it was possible that because there is no derivative at x = 5, there could also possibly be or not be a point of inflection there, too. The book says it's "possibly" a point of inflection, but going by the definition, we just presumed it to "definitely" be one.