1. ## 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?

2. We know that the minimum is attained at a point $\displaystyle c\in \left[a,b\right]$. We only have to show that it can't be at $\displaystyle a$ or $\displaystyle b$. If the minimum is attained at $\displaystyle a$ for example then exists $\displaystyle h_0>0$ such that if $\displaystyle 0<h<h_0$ then $\displaystyle \frac{f(a+h)-f(a)}h<0$. Now, don't forgive that $\displaystyle h$ is positive in order to find a contradiction.

3. Originally Posted by girdav
$\displaystyle \frac{f(a+h)-f(a)}h<0$
Is this inequality the wrong way round? Should it not be >?

4. Originally Posted by worc3247
Is this inequality the wrong way round? Should it not be >?
No, since $\displaystyle f'(a)<0$. Then I use the definition of a limit.

5. Ok thanks

6. ## Re: Continuity/Differentiability Question

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