Q:

Does unique maximum at interior point imply negative second derivative?

Suppose that a function $f$ is twice continuously differentiable on an open interval $(a,b)$, and also attains a unique maximum at some $p \in (a,b)$. Is it true that $f''(p) < 0$? If so, why? If not, feel free to assume infinite differentiabilty of $f$. 
I know how to prove the other direction, but this direction is confusing me a bit. Any help?

A:

Look at the curve
$y(x) = -x^4 \tag 1$
for
$x \in (-1, 1); \tag 2$
then 
$y(0) = 0 \tag 3$
is the unique maximum, but since
$y''(x) = -12x^2, \tag 4$
$y''(0) = 0. \tag 5$
So the answer appears to be "no".

