91Ó°ÊÓ

$Tisatrianglea,b,careitssides.$

$s=\frac{1}{2}\left(a+b+c\right)andA=\sqrt{s(s-a)(s-b)(s-c)}.$

The perimeter is the sum of all sides, which is $a+b+c=2s.$

Therefore the constraint function:

The function which maximize the area is:

By the method of Lagrange's multiplier,

Now whatever be the value of $\mathrm{λ}$, all the components of $∇f$must be same.

So,

$$\begin{gathered} (s-b)(s-c)=(s-a)(s-c)=(s-a)(s-b). \\ Thisimplies, \\ s-a=s-b=s-c, \\ And, \\ a=b=c. \end{gathered}$$

Hence, it is proved that the triangle must be equilateral in order to have its area maximum.