Here's a problem that should be intuitive, yet I cannot seem to get at a proof.
Let $f(x)$ be a function $ \in C^\infty $ such that $f'(a)<0$ and $f'(b)>0$ for $ a < b $. Show that f must have a local minimum.
Clearly, by the IVT, I can find a point $ x\in[a,b]$ where $f'(x)=0$.
My idea is something along the lines of:
If $f''(x)< 0$, then we're done
If $f''(x)\geq 0$, then there will have to be another point, $x_1$ where $f'(x_1)=0$ and if $f''(x_1)<0$ we're done and if not then... and that this cannot go on forever.