Let $f$ be a real function which is differentiable on some interval $[0,a]$ ($a>0$) that satisfies :
- $f'$ is continuous on $[0,a]$
- $f'(0)=f(0)=0$
- $f(a).f'(a)<0$
It is asked to prove that $f'$ vanishes at some point $c\in (0,a)$.
My approach :
Without loss of generality (working with $-f$ to address the other case) Assume that $f(a)<0 $ and $f'(a)>0$.
Claim : there exists $c \in (0,a)$ such that $f(c) <f(a)$ (because otherwise $f'_g(a) \le 0$)
$f$ is continuous on $[0,c]$ thus $f$ takes the (intermediate) value $f(a)$ at some $d\in (0,c)$. We can then apply Rolle's theorem to $f$ on $[d,a]$ to get the sought result.
Comment :
It seems the conditions : $f'(0) = 0$ and the continuity of $f'$ on $[0,a]$ are unnecessary.
Any remarks, rectifications are welcome. Thanks.