Definitely Maybe: Concavity and Point of Inflection

[dozzer]

Homework Statement

f is a continuous function on [0, 8] and satisfies the following:
Second Derivative Sign Test
x ; f''
0 $$\leq$$ x < 3 ; -
3 ; 0
3 < x < 5 ; +
5 ; Does Not Exist
5 < x < 6 ; -
6 ; 0
6 < x $$\leq$$ 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.

 

[clamtrox]

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

Can you think how this would reflect on the question? Can you think of two examples to satisfy the conditions on f''(x), one that does not have a tangent at x=5 and one which does?