Is it true that $\text{min}\{x/2, y/2\} = \frac{1}{2}\text{min}\{x, y\}$? Is it true that $\text{min}\{x/2, y/2\} = \frac{1}{2}\text{min}\{x, y\}$? Or more generally, $\text{min}\{cx, cy\} = c \cdot \text{min}\{x, y\}$?
I think that the answer is YES. But, I got this guess by just plugging in a few numbers. Maybe I am missing some sort of clever counterexample. Can someone please help me?
 A: Verify the following formula for the min : 
$$
\min\{x,y\} = \frac{x+y}{2} - \left|\frac {x-y}2\right|
$$
To see this , check the two cases $x \geq y$ and $x < y$.
Now, substitute $c\geq 0$ into the right hand side. Can you see what happens? Finally, see why $c<0$ fails.

Use the fact that $x+y = \max\{x,y\} +\max\{x,y\}$ to obtain an expression for $\max\{x,y\}$ and verify a similar property for the maximum. Also, you can "iterate" the formula for $\max\{a_1,a_2,...,a_n\}$ to see that a similar property would hold true here as well.
A: The most general question is: take any function $f$, when is $\min\{f(x),f(y)\} = f(\min\{x,y\})$ for all $x$ and $y$?
To analyze this. The insight you need is that $\min\{x,y\}$ is either $x$ or $y$ depending on whether or not $x$ or $y$ is smaller. This is obvious but look at what it gets us. Suppose $x \le y$. Then $f(\min\{x,y\}) = f(x)$. Similarly, if $y \le x$ then $f(\min\{x,y\}) = f(y)$.
So now the question is: if $x \le y$, is $f(x) = \min\{f(x),f(y)\}$ and if $y \le x$, is $f(y) = \min\{f(x),f(y)\}$?
See if you can write down a simple criterion $f$ needs to satisfy in order to make this work.
Can you work out a criterion in which $x$ and $y$ are always swapped? I.e. if $f(\min\{x,y\}) = f(x)$ then $\min\{f(x),f(y)\} = f(y)$ and if $f(\min\{x,y\}) = f(y)$ then $\min\{f(x),f(y)\} = f(x)$.
A: The counterexample is $c=0$ or $c < 0$.
But if $c > 0,$ then for any set $A$ of real numbers, if $a, b \in A$ then 

$a < b \iff ca < cb$.

So if $k= \min A$ exists,  then $\min \{ca|a \in A\}$ exists and is $ca$  (because $a \le b$ for all $b \in A$ if and only if $ca \le cb$ for all $cb \in  \{ca|a \in A\}$).
