What is the $\min(x+y)$ if $xy=k$ and $x, y, k \in Z$. I've got this question from thinking of the minimum perimeter of a given area of a rectangle. Here's my try:
From AM-GM: $x+y\le 2 \sqrt {xy}$
$\Rightarrow x+y\le \lceil 2\sqrt {k} \rceil$
$\Rightarrow x+y =\le \lceil 2\sqrt {k} \rceil$
From Veita's formula:
$t^2 -(\lceil 2\sqrt {k} \rceil)t +k$
And finally by the Quadratic formula:
$t=\frac {\lceil 2\sqrt {k} \rceil \pm \sqrt {(\lceil 2\sqrt {k} \rceil)^2 -4k}} {2}$
The Formula that I've got works fine with lots of numbers but not all of them (e. g: $k=5$ spits (3.618..., 1.381...)). Can you help me work this out? Thanks in advance.
You cannot obtain minimum perimeter. If you do it will be just a straight line with any of the $x,y=0$.
Where as you can obtain the Maximum perimeter by using the concept of derivatives. If you are interested to find this out, then let me know.