$\displaystyle f(x) : [-1, \; 1]\to\mathbb{R}$ is differentiable on $\displaystyle (-1, \; 1)$ and continuous on $\displaystyle [-1, \; 1]$.

Furthermore, $\displaystyle f(x)$ satisfies following condition :

$\displaystyle f'(x)\ge-1\;\;\forall x\in (-1, 1)\\ f(-1)\ge f(1)\\ \int_{-1}^{1}f(x)\mathrm{d}x = 0.$

Then, prove $\displaystyle \int_{-1}^{1}(f(x))^{2}{d}x\le\frac{2}{3}$

Please help me.