I have a homework question which is:
If $f : [a,b]->R$ is continuous in $[a,b]$ and differentiable at $(a,b)$ and exists a point $c$ in $(a,b)$ such that $(f(c)-f(a))(f(b)-f(c))<0$ then prove that there is a point $d$ in $(a,b)$ such that f'(d)=0.
I am having trouble proving this - I am sure I am missing some simple algebra trick to show that $\frac {f(b)-f(a)}{b-a}=0$ or something like that...
Can someone help me please?
Thanks :)