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