Properties of min(x,y) and max(x,y) operators

Question

Is $\min(x^2,y^2)=[\min(x,y)]^2$, and similarly for $\max(x,y)$?

Also, is $\sqrt{\min(x^2,y^2)}=\min(x,y)$? Do other non-linear operations work?

In general, what are the other interesting properties of these operators, and where can I study more about them?

How about if we take, $x=-3,y=-4$ – Dinesh Jul 12 '20 at 15:59 — Dinesh, Jul 12 '20 at 15:59

score 2 · Accepted Answer · answered Jul 12 '20 at 16:08

2

No. $\min\{2^2,(-3)^2\}=2^2\ne\min\{2,-3\}^2=(-3)^2$. You can see even $\max$ will not work for this example.

In general, it will work when $x\le y\implies x^2\le y^2$, i.e. in the monotonically increasing section of the curve $y=x^2$, where $x,y\ge0$.

answered Jul 12 '20 at 16:08

Shubham Johri

Thanks, any resources where I can read up more on this? – PGupta Jul 12 '20 at 16:15

score 2 · Answer 2 · answered Jul 12 '20 at 16:43

2

These are related to the absolute value:

$$\text{max}(x,y) = \frac{1}{2}(x + y + |x - y| ),$$

$$\text{min}(x,y) = \frac{1}{2}(x + y - |x - y| ),$$

and are thus continuous functions of $(x,y)$.

answered Jul 12 '20 at 16:43

James

2 Answers2