Finding Triangle with constant perimeter and largest area (Lagrange Multiplier)

Question

Question is to find Finding Triangle with constant perimeter and largest area by method of lagrange multiplier .

What i have done is that i have firstly taken $x+y+z=2k$ , where x,y,z are sides of triangle..k is any constant

Then i use Heron's formula as $\sqrt{s(s-x)(s-y)(s-z)}$ , where $s = (x +y+z)/2$ ....

Since Area = $\sqrt{s(s-x)(s-y)(s-z)}$

So substituting values of $s$ and replacing $z$ by $2k-x-y$ (to make it to two variable problem ) i finally get

$f(x,y) = k(k-x)(k-y)(x+y-k)$ ...($A$ is squared so as to be easy easy derivatives)

And my constraint equation is $g=x+y+z-2k=0 $ .....But problem here is that constraint consists z also . So i feel stuck to use LAGRANGE MULTIPLIER Method.....Can any1 help me furthure what to do from here .THANKS

Answer 1

There is no reason to eliminate $z$. The constraint is $x+y+z=2k$, and the objective function is $(k-x)(k-y)(k-z)$.