Let $f$ be a continuous function of one variable. Show that if $f$ has two local maxima, then $f$ must also have a local minimum.
Let $x_1,x_2$ be the points where $f$ attains the maxima; If $x_1<x_2$, suppose that $f(x_1)<f(x_2)$. Since $f$ is continuous there must exist a point $c$,between $x_1$ and $x_2$, such that $f(c)=f(x_1)$. By Rolle's theorem there is a point(between $x_1$ and $c$), say $x_3$, such that $f'(x_3)=0$; So how could I show that $f''(x_3)>0$?