TITLE: If $f$ is an infinitely differentiable function then does this hold?
QUESTION [11 upvotes]: If $f$ is differentiable infinite number of times, then there exist $x\in (0,1)$ such that
$$\frac{f(1)-f(x)}{x}=f'(x)$$.
I have tried to use Lagranges Mean Value Theorem .
It lands me with $f(1)-f'(x)=(1-x)f'(x+\theta(1-x))$ where $\theta\in(0,1)$.
Then I thought that I could differentiate both sides again and again to get some form of $\theta^{n}$ but to no avail.
Even the differential equation $xf'(x)+f(x)=f(1)$ admits a solution but I don't think that it will help me.
Can anyone tell me how to proceed with this?

REPLY [2 votes]: To extend on @Gribouillis's solution.
Take $g(x)=xf(x)-xf(1)$ . Then $g(0)=g(1)=0$.
Then there exist $c\in(0,1)$ such that $g'(c)=0$.
$g'(c)=f(c)+cf'(c)-f(1)=0\implies \frac{f(1)-f(c)}{c}=f'(c)$ . Thus you have your answer. And yes for the intuition regarding this, the solution of the ode was a good observation.