real analysis – Prove $f(1)-f(-1)>f’left(-frac{1}{sqrt{3}}right)+f’left(frac{1}{sqrt{3}}right)$.

Suppose the 5th order derivative function of $f(x)$ ,say, $f^{(5)}(x)>0$ for $xin (-1,1)$.Prove $f(1)-f(-1)>f’left(-dfrac{1}{sqrt{3}}right)+f’left(dfrac{1}{sqrt{3}}right)$.

I know this can be done directly by Gauss-Legendre formula, but does there exist a more elementary proof?