When is the sum of a non-convex function with a quadratic function convex?

Question

I am looking for conditions on parameters $a$ and $b$ and on non-convex function $g$ such that the scalar function

$$f(x) = a g(x) + \frac b2 x^2$$

is convex. It would be a great help if someone can guide me on this problem.

Answer 1

If you are willing to (or can always) choose $a>0$, and $g$ is twice differentiable on an open interval, then you obtain your result on that interval if $$g''>-b.$$