maximal point of a combination of continuous sub-additive monotone function

Question

Let $f:\mathbb{R}\to \mathbb{R}$ be a monotonically increasing sub-additive continous function. Let $0<t\in \mathbb{R}$. Is $\frac{t}{2}$ a local maximum of the function $f\left( x \right) + f\left( t-x \right)$

Answer 1

Let $$f(x)=\max\{2x,x+1\}=\begin{cases}2x,&x\ge1\\x+1,&x\le 1\end{cases}.$$ Being increasing, continuous, or subadditive is preserved under $\max$, and funxtions $x\mapsto ax+b$ with $a>0$ and $b\ge 0$ are increasing, continuous, and subadditive. Hence, $f$ is an increasing, continuous, subadditive function. With $t=2$, we consider the function $$f(x)+f(2-x)= \begin{cases}2x+(2-x)+1,&x\ge1\\(x+1)+2(2-x),&x\le 1\end{cases}=\begin{cases}x+3,&x\ge1\\5-x,&x\le 1\end{cases}$$ This has a global strict minimum at $\frac t2=1$.