Suppose $f$ is differentiable with $f'(x)>0\;\forall x$ and $f(1)=0$ how can I prove that $F(1)<0$?

Question

Suppose $f$ is differentiable with $f'(x)>0\;\forall x$ and $f(1)=0$ how can I prove that $F(1)<0$?

where $F(x)=\int_0^x f(t)dt$ (the primitive integral)

also the function $F$ is tw0-times differentiable?