take The more example such $f''(x)\ge 0,x\ge 0,f(0)=0$

Question

Which functions satisfy $$f''(x)\ge 0,x\ge 0,f(0)=0$$

take The more example and the better. Thank you for providing a function that satisfies these conditions.

I have found this $f(x)=x^p(p>1)$

can you have more? Thanks

Answer 1

Consider any convex function that is twice differentiable, $g$, let $f(x) = g(x) - g(0)$ .

Then we have $f''(x) = g''(x) \ge 0$ and $f(0)=g(0)-g(0)=0$.