Question

How do I find the largest value of the product x*y given that x and y are related by x + y = 4?

Answer 1

The sum of the quantities being given the product is maximum when they are equal,so here x=y=2 Hence, 4

Answer 2

Isn't there some derivative or something that I have to use?

Answer 3

FoolForMath is correct, though I can't prove it without calculus. Oh this is calculus? ok then
x+y=4
y=4-x
xy=x(4-x)=4x-x^2=f(x)
f'(x)=0=4-2x
x=2

Answer 4

Maxima-minima approach is really a overkill here,it's a trivial AM-GM inequality application.

Answer 5

@TuringTest:Read on here: http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means and really avoid using calculus in these kind of trivial problems :)

Answer 6

yes, but if he is in calculus we need to do the problem this way. Of course I know areas max out when L=w, I just wanted to prove it.

Answer 7

it's not a difficult derivative to do and proving the idea using geometry would be far more difficult. You are right that my proof is technically incomplete though, so yes
f''(x)=-2<0
hence the graph is concave down and (2,2) is a maximum

Answer 8

Also,your prove is incomplete without the second derivative test for proving maxima i.e f''(x) < 0,concave down,Hence Maxima! :)

Answer 9

Due to server fault,my post got delayed I was trying to correct "with" to "without"

Answer 10

Um I don't understand what you meant by "proving the idea using geometry would be far more difficult." It's actually easy to see and apply things geometrically :)

Answer 11

It may be easy to see when you look at it, but as far as technically proving the general case for an n-dimensional object or something I don't know an easy way to do it with geometry. If you have a link on an easy proof for maximization of an area without calculus let me know. The one on the wiki page looks more confusing to me than the derivative test.

Answer 12

Parabola -x^2+4x has max at (2,4), so x=2 and y=2 without any complications of calculus.

Answer 13

Actually I guess I understand the proof on wikipedia, but I still say it's important to know the calculus method.