Let $f(x)$ be a continuous function in $[0,1]$ and differentiable in $(0,1)$ such that $f(1) = 0$
Prove that there is $x_0\in(0,1)$ such that $x_0\cdot f'(x_0) + f(x_0) = 0$
My attempt -
I really couldn't make much progress but it does feel to me like some kind of Lagrange Theorem manipulation. i tried some kind of messing around, and could only prove that there exists $x_0$ such that - $$f'(x_0) = -f(0)$$
Any hints ?