# Find the “sweetspot” of a subadditive function

Please excuse my sloppy notation since I am not an expert. However, I stumbled upon the following problem and I would be very grateful for any help.

Assume we are given a continuous, monotonically increasing and subadditive function $$f:[a,b]\rightarrow \mathbb{R}$$. Say a "sweetspot" is a value $$x\in[a,b]$$ minimizing $$\max\{x-a,f(b) - f(x)\}$$. Are there general concepts to calculate such a "sweetspot"? How is this problem called in general?

• Since $x-a$ is increasing and $f(b)-f(x)$ is decreasing, the optimum is attained when the two are equal and can be found through bisection search, can't it? – Rahul Sep 11 at 10:20
• I concur with @Rahul. This problem is equivalent to solving $x - a = f(b) - f(x)$, an equation that should have one solution. Any method for solving equations of the form $g(x) = c$ will work, but in general, this is not possible to do analytically. – Theo Bendit Sep 11 at 10:38