I'm working about this :
Let $x \geq 0$ and $f(x)$ be a twice differentiable and continuous function such that : $$f''(x)^2\leq f'(x)+x^2$$ With $f(0)=0$ and $f'(0)=0$
Prove that $f(x)^2\leq \frac{x^3}{4 \pi^2}(f(x)+\frac{x^3}{3})$
My friend tells me that we can use the Wirtinger's inequality but I don't see how...
Can someone help me ?
Thanks in advance for your time