Monotonicity of the sum/product/max of two monotone functions

Question

Suppose two monotone functions $f$ and $g$ (both weakly increasing or both weakly decreasing) are given. How can it be shown that $f+g, f \cdot g, \max(f,g)$ is again monotone (either weakly increasing or weakly decreasing)? Is there a reference to a text book?

Answer 1

Let $f,g$ bi monotonuous increasing functions on the domain $D$.

Then for all $x,y \in D$ with $x \leq y$ we know that $f(x)\leq f(y)$ and $g(x) \leq g(x)$.

If we add those inequalities, we get $f(x)+g(x) \leq f(y)+g(y)$ which is equivalent to $(f+g)(x) \leq (f+g)(y)$.

You can use this for the other two exercises.