Inequality with extreme values

Question

$f:[0,1]\rightarrow \mathbb{R}$ is continuous and non-constant function. $F$ is the indefinite integral of $f$ such that $F(0)=F(1)=0$. $m$ and $M$ are the minimum and the maximum values of $f$. Now we define $g:[0,1]\rightarrow \mathbb{R},\: g(x)=xf(x)$ and the indefinite integral of $g$ is $G$, such that $G(0)=0$. Prove that $|G(1)|\leq\dfrac{−mM}{2(M−m)}$ . Can somebody help me,please?