I know this is the wrong forum, but please help me.
It is about a simple looking math problem.

A = 2xy / (x^2 + y^2)
x>0, y>0
What is the maximum of 'A'?

I'm stuck with this. I'd appreciate anyone who can show the solution.


Winter vacation has ended, and the university enterence test is at November 2013.... I sadly have terribly short time for Go..

It is 1. There is a common inequality (called arithmetic-geometric mean inequality, but nevermind that) that says that

x^2 + y^2 >= 2xy.

to prove it, you can write x^2 + y^2 - 2xy = (x-y)^2 >= 0

For an alternative geometric solution, you could draw a circle and transform to polar coordinates:
x = r sin(t), y = r cos(t), 2xy / (x^2 + r^2) = (some trig manipulation) = sin(2t)

If you've done calculus, a possible thrid approach is to take the partial derivative of A w.r.t. x or y, and set it to 0.

This a lot less elegant than the solutions posted so far, though.

Specifically:

dA/Dy=(2x^3-2xy^2)/(x^2+y^2)^2.

Setting this to 0, and using x>0 and y>0, gets x^2=y^2. Hence x=y at the maximum, where A=1. (There's a little more work to do to show this a maximum not a minimum.)

Real thanks for everyone who replied (and will reply...) to my post.
And... let me commit my second abuse to this forum.

The (math) problem, below in a italic font, that I'm again in trouble with wants me to find the truth values of propositions about matrix. The original problem is written in Korean, so I may have used some improper mathematical expressions in English.

'A' is a 2x2 matrix, 'O' is a 2x2 zero matrix, 'n' ∈ a set of all natural numbers

(1)
A^n = O → A^2 = O

(2)
A 2x2 matrix 'X' that 'X^2 = A' always exists regardless of the value of 'A'. (Whatever value 'A' has, there must be some kind of a 2x2 matrix 'X' that 'X^2 = A')

(3)
'A' that 'A^2 = ( 1 2 )' exists.
____________( 3 4 )

I'd again, of course, appreciate anyone who can show the solution.

You have an oops!

OK. What must be true of a, b, c, and d for A^2 to be 0?

a=b=c=d=0 or something like a=1, b=1, c=-1, d=-1 (some cases when ad-bc = 0)

A <> 0, A^2 = 0 , where A is a 2x2 matrix.

Write A using 2 real variables.

----

Given: A is a 2x2 matrix.
Prove: A^4 = 0 -> A^2 = 0.