If we have a function f which is continuous and differentiable on $[a,b]$ and $(a,b)$ respectively. It is given that there exists a $c$ in $(a,b)$ such that $$(f(c) - f(a))(f(b)-f(c)) < 0$$ We have to show that there is a $t\in(a,b)$ such that $f'(t) = 0$.
My thoughts so far: We have a case where $f(c)-f(a)$ is negative meaning $f(b)-f(c)$ is positive. This means $f(c) < f(a)$ and $f(c)>f(b)$. But this doesn't seem to imply that there is a turning point, so I'm stuck. Any help would be appreciated.