Let $g$ be differentiable on $[0, \infty)$ with $g(x) \geq 0$ for all $x \geq 0$ with $g(0) = 0$. Moreover, suppose $g'(x) \geq f'(x)$ for all $x\geq 0$ where $f(x) = x^2$. Also suppose $f$ and $g$ are distinct functions.
How can I find the minimum possible value of $g(4)$?
I know we require $g'(x) \geq 2x$ for all $x \geq 0$. I'm thinking that this has to do with integration. I'm not sure how integration works with bounds because I don't think we can just integrate both sides of $g'(x) \geq f'(x)$ to get $g(x) \geq f(x)$ (or can we?). If this is allowed, then the problem's really easy: the answer would just be $g(4) \geq f(4) = 16$.
Any help is appreciated