Proving a maximum and minimum inequality

Question

I'm trying to show that $\min{(x,y)} \le \frac{a}{a+b}\cdot x + \frac{b}{a+b} \cdot y \le \max{(x,y)}$, for all $a$, $b$, $x$, $y$ $>0$.

Any ideas?

Answer 1

$x,y\ge\min(x,y)\implies\frac{ax+by}{a+b}\ge\frac{a\min(x,y)+b\min(x,y)}{a+b}=\min(x,y)~~\because a,b>0$

Similarly use the fact $x,y\le\max(x,y)$ to prove the other part.