Problem : let $f:[0,\infty)\to\mathbb{R}$ be continuous on $[0,\infty)$, differentiable on $(0,\infty)$, Given that $f(0)=0$, and $\lim\limits_{x\to\infty}f(x)=0$. Prove that there is $c\in (0,\infty)$ such that f'(c)=0.
I need help on the above problem! Thanks. Also let me know if my proof provided below is true, if not please let me know the right answer.
Here is my first tentative: let $a>0$. Since $f$ is continuous and differentiable on $(0,\infty)$ and hence on $(0, a)$, then using the Mean Value Theorem we conclude that there is $b\in(0,a)$ such that: f(a)-f(0)= f'(b)(a-0). as $b$ tends to $\infty$, $f(a)$ tends to $0$ which makes the left hand side of the equality equal to $0$, and this implies f'(b)=0.