Among all triangles with a fixed perimeter p a b c show t

Among all triangles with a fixed perimeter p = a + b + c, show that equilateral triangle has the greatest area. You may want to use the formula of Heron A= squarerootof s(s - a)(s - b)(s - c) for the area A of a triangle in terms of the lengths a, I. and Cui the sides, in terms of half the perimeter s = p/2. Find the point (x, y, z) in the first octant that is on the surface given xy^2z^4 = 1/4 and is closest to the origin (0, 0, 0).

Solution

A= sqrt{(s(s-a)(s-b)(s-c)}

s = (a+b+c)/2

compute derivatives w.r.t. a, b and c, but since they are not independent variables, we first substitute the semi-perimeter equation into the are equation, to get rid of c.

A = sqrt[(s(s-a)(s-b)(a+b -s)

find the derivative w.r.t . a:

dA/da = [s(s-b)(2s- 2a -b)/2A

dA/da =0;   [s(s-b)(2s- 2a -b)/2A =0

Neglect solution as it would make area =0 : s = b

2s - 2a -b =0

Find dA/db =0 ; we get 2s -2b -a =0

we have : 2s - 2a -b =0 ----(1)

2s -2b -a =0 ----(2)

solve the two equations : a = b = 2s/3

c = 2s/3

which means : a= b = c = 2s/3

It is am equialateral triangle with maximum area