# real analysis – Show that \$int^b_a f^3(x)dx le ( int^b_a f(x)dx )^2\$ when \$f(a)=0\$ and \$0 le f'(x) le 1\$

Assume that $$f:(a,b) to mathbb R$$ is a continuously differentiable function. We want to show that $$int^b_a f^3(x)dx le ( int^b_a f(x)dx )^2$$ if we have $$f(a)=0$$ and $$0 le f'(x) le 1$$ for all $$x in (a,b)$$.

My attempt:

I don’t know if my approach is correct, but I started by changing the upper limits of the integrals to a variable $$y$$ so that I can differentiate and obtain $$f’$$. After changing the upper limits, I want to show

$$( int^y_a f(x)dx )^2 – int^y_a f^3(x)dx ge 0$$

for $$y in (a,b)$$. This holds for $$y=a$$. So I will now show that the derivative of LHS w.r.t. $$y$$ is positive. That is, I want to show

$$2f(y)(int^y_a f(x)dx) – f^3(y) ge 0$$

Again, this holds for $$y=a$$ since $$f(a)=0$$. And I take the derivative again and want to show that it is positive. The derivative is

$$2f'(y)(int^y_a f(x)dx) +2f^2(y)-3f^2(y)f'(y)$$

I couldn’t get rid of the integral and I am stuck here. Can you help me by proposing another approach or giving a solution?