Prove sum of two convex funtions is convex using first and second derivatives

Question

Consider if I have two functions f(x) and g(x) which are both convex. How do I prove that their sum is also convex using the first and second derivatives?. I know if the second derivative of a function is positive then it is convex but I'm not sure how to prove this for a general case like this.

Answer 1

Since you're talking about second derivatives, I'll assume $f$ and $g$ are twice differentiable.

Then $f, g$ are convex if their second derivatives are nonnegative. But then the second derivative of $f+g$ is $f''+g'' \geq 0$ since $f'' \geq 0, g'' \geq 0$.

So $f+g$ is convex, since it has nonnegative second derivative.