# Lagrange Multiplier

• Nov 8th 2009, 06:24 PM
LostMathMan
Lagrange Multiplier
It seems relatively simple but I still can't get it figure out.

Use lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.

$\displaystyle A=\sqrt(s(s-x)(s-y)(s-z))$ where $\displaystyle s=p/2$ and x, y, z are the lengths of the sides.
• Nov 9th 2009, 12:08 AM
Opalg
Quote:

Originally Posted by LostMathMan
It seems relatively simple but I still can't get it figure out.

Use lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.

$\displaystyle A=\sqrt(s(s-x)(s-y)(s-z))$ where $\displaystyle s=p/2$ and x, y, z are the lengths of the sides.

It's easier to maximise A^2 than A. So the problem is to maximise s(s–x)(s–y)(s–z) subject to the constraint x+y+z=2s, where s is constant. (There's also the condition that all the quantities x, y, z, s–x, s–y, s–z should be positive.) That should be a pretty standard Lagrange multiplier problem.