Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square.

Let the sides of the rectangle be x and y and let fand g represent the area (A) and perimeter (p), respectively. Find the following.

A f(x, y) -

p = g(x, y)-

Vf(x, y)-

Then y- implies that x .

Therefore, the rectangle with maximum area is a square with side length.

Step-by-step explanation:

1. Method of Langrage Multipliers:

To find the extremum values of f(x,y) subject to constraint g(x,y) = k

step1;

find all values of x,y and λ, such that :

                                 Δλ(f,x) = λΔg(x,y)

                           And

                                g(x,y) = k

2. let the two side of the rectangle be x and y

therefore

f(x,y) = xy And g(x,y)= 2(x+y)=p

fₓ=λgₓ => y=2λ 1

fy = λgy => x = 2λ2

using equation and 1 and 2

λ=0, but this is not possible because tis implies x = y = 0, which gives  0 perimeter

or

x=y

Hence the rectangle must be a square

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