Suppose we have two functions $f(x) = a \sqrt{x}+b\cdot x$ and $g(x)$ which can be any concave function, and $x\geq 0$ for both functions $f(x)$ and $g(x)$. Is it possible for us to compute a solution for $a$ and $b$ (I think the solution depends on function $g$ here) such that $$f(x) \geq g(x) - \delta, \forall x\geq 0$$ where $\delta$ is a non-negative constant.
My thought for this is to rewrite the requirement $f(x) \geq g(x) - \delta$ as $\delta \geq g(x) - f(x)$. Define $h(x) = g(x) - f(x)$, then we only need to maximize $h(x)$ for $x\geq0$ and let the maximum of $h(x)$ be less than $\delta$. However, $h(x)$ here is the difference between two concave functions, which should be NP-Hard to maximize in general.
Any other thoughts on finding a solution for $a$ and $b$ here?