A problem asks the following
$f$ is a twice-differentiable function on some segment $[a,b]$ such that $f(a)=f(b)$ and $f'(a)f'(b)<0$. it asks to prove that the second derivative of $f$ vanishes at some point between $a$ and $b$ (strictly).
What about this situation
You are right: the statement is false and you have the right idea. An example would be $f(x)=x^2$, $a=-1$, and $b=1$. Then $f'(a)f'(b)=-4<0$, but you always have $f''(x)=2>0$.