Help me to solve this following problem. Thanks a lot.

"Let f(x) is a continuous function $\displaystyle f:[0,1]\to [0,+\infty)$. Put $\displaystyle g(x)=1+2\int_0^x f(x)$. Assume that $\displaystyle g(x)\ge (f(x))^2 \forall x\in[0,1] $. Prove that $\displaystyle g(x)\le (1+x)^2$"