Continuous and Differentiable Function has a Local Maximum

Question

If $f$ is a differentiable function such that $f(a)=f(b)=0$, and there exists $c \in (a, b)$ such that $f(c)>0$, prove that $f$ has a positive local maximum.

Answer 1

Find a $\delta>0$ small enough such that $c\in(a+\delta,b-\delta)$. Since $f$ is continuous on $S=[a+\delta,b-\delta]$, $f$ attains its maximum on $S$, so $f(x_{0})=\max_{x\in[a+\delta,b-\delta]}f(x)$ for some $x_{0}\in[a+\delta,b-\delta]$, and we have $f(x_{0})\geq f(c)>0$.