   Chapter 14.8, Problem 30E

Chapter
Section
Textbook Problem

Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.Hint: Use Heron’s formula for the area: A   =   s ( s   −   x ) ( s   −   y ) ( s   −   z ) where s = p/2 and x, y, z are the lengths of the sides.

To determine

To find: The maximum area of a triangle subject to the perimeter by using Lagrange multipliers.

Explanation

Given:

The Heron’s formula for the area is A=s(sx)(sy)(sz) where s=p2 and x, y, z are the lengths of the sides.

The perimeter of a triangle is p=x+y+z .

Definition used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx fy=λgy fz=λgz and g(x,y,z)=k ”.

Calculation:

Let the given function of f(x,y,z)=s(sx)(sy)(sz) .

The given function g(x,y,z)=x+y+z .

The Lagrange multipliers f(x,y,z)=λg(x,y,z) is computed as follows,

f(x,y,z)=λg(x,y,z)fx,fy,fz=λgx,gy,gzfx(s(sx)(sy)(sz)),fy(s(sx)(sy)(sz)),fz(s(sx)(sy)(sz))=λgx(x+y+z),gy(x+y+z),gz(x+y+z)(s(1)(sy)(sz)),(s(sx)(1)(sz)),(s(sx)(y)(1))=λ1,1,1(s(sy)(sz)),(s(sx)(s

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