Continuity/Differentiability Question

Suppose $\displaystyle f:[a,b] \rightarrow \mathbb{R}$ is differentiable on [a,b] (with one sided derivatives at the end points). Show that if $\displaystyle f\prime(a)<0<f\prime(b)$ then the minimum of f is attained at a point $\displaystyle c \in (a,b)$. Note: You may not assume the derivative of f is continuous.

My first thoughts was just to use IVT on the derivative until I saw the note, now I'm not sure what to do. Any help where to start?

Re: Continuity/Differentiability Question

And just so you know, the IVT property also holds for derivatives, regardless of differentiability.