# Prove the equation has a root.

Assume that $f$ is a bounded and differentiable function in $(0,1)$. If $f({1\over 2})=0$, prove that the equation, $$2f(x)+xf'(x)=0,$$ has at least one root in $(0,{{1}\over{2}})$.

I tried to do it using Rolle's Theorem. Because the left side of the equation looks like the derivative of some function. And if I find a function $F$ s.t. $F'(x)=2f(x)+xf(x)$ and $F(0)=F({1\over 2})$, then I can use Rolle's Theorem to get the conclusion. I've found $F$, which is $$F(x)=\int_{0}^{x}f(t)\,dt+xf(x),$$ s.t. $F'(x)=2f(x)+xf(x)$, but the only problem is that I can't ask for $F(0)=F({1\over 2})$.

Hint: What's the derivative of $F(x)=x^2f(x)$?
@CityOfGod: Apply Rolle's theorem to the $F(x)$ in my answer. –  Jyrki Lahtonen Feb 15 '13 at 21:53