Is this situation possible? Impossible?

Question

Consider the interval $[0,a],a>0$ and $c\in[0,a]$. Suppose that each function $f$ is continuous over $[0,a]$:

$f$ has two maxima, two minima and a unique value $c$ such that $f'(c)=0$

How do I approach the problem? Which theorems do I apply?

Answer 1

It is possible. You may be tempted to think that you must have the derivative zero at the local maxima and minima, but we are not given that the function is differentiable everywhere, just that it is differentiable at $c$. An example is below.

If the function were specified to be differentiable over the whole interval it would be impossible because the derivative would have to be zero at $A$ as well.

Answer 2

It is probably a typo, because $[a,a]=\{a\}$. Any function defined here cannot have two minima...

As a comment suggests, it may be $[0,a]$.