0
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ Consider $g(x) = e^x f(x)$. $\endgroup$ – John Hughes Oct 14 '17 at 17:11
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.