Suppose f and g are two increasing convex functions on [0,1] with $\displaystyle f(0) \leq g(0)$ and $\displaystyle f(1)=g(1)=1$. Show that

$\displaystyle (g'(x)-f'(x))(1-x) + g(x) - f(x) \geq 0$.

I know that $\displaystyle f(x)+(1-x)f'(x) \leq 1$, with the same for g. But that doesn't look that helpfull.