# Derivative of a point equal to its output in a function that's continuous in a domain.

Assume $f$ is a function such that $f(a)=f(b)=0$.

The function is defined and continuous on the domain $[a,b]$ and is differentiable on $(a,b)$.

Prove that there is a point $c$ in that domain such that $f(c)=f'(c)$.

I don't know how to solve this problem, I wanted to approach this by assuming that there is a number of possible continuous functions that belong to the domain and at least one of them has a derivative that's equal to the output at point $c$. Other than that, I'm clueless, please help me out!

• Consider $g(x) = e^x f(x)$. – John Hughes Oct 14 '17 at 17:11

## 1 Answer

Let $g (x)=e^{-x}f (x)$.

$g$ is continuous at $[a,b]$ with $g (a)=g (b)=0$.

$g$ is differentiable at $(a,b)$.

By Rolle's Theorem

$$\exists c\in (a,b) \;\;: g'(c)=0$$

but $g'(c)=\Bigl(f'(c)-f (c)\Bigr)e^{-c}$.

You can finish.