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Chapter 12: Q. 57 (page 986)

Prove that a square maximizes the area of all rectangles with perimeter P.

Short Answer

Expert verified

∇g=2,2∇f=b,aSolving,∇f=λ∇gweget,a=bwhichistheconditionoftherectangletobesquare.

Step by step solution

01

Step 1. Given Information.

Given a rectangle with perimeter P. Let a and b be the dimensions of the rectangle.

02

Step 2. Finding the constraint.

The perimeter is the sum of all sides, which is 2a+2b.

Therefore, the constraint function is:

g(a,b)=2a+2b.

and it's gradient is:

∇g=2,2.

The function which maximize the area is:

f(a,b)=A=ab.

and it's gradient is:

∇f=b,a.

03

Step 3. Using Lagrange's multiplier.

By the method of Lagrange's multiplier, ∇f=λ∇g,So,∇f=λ2,2=2λ,2λ.

Now whatever be the value of λ, all the components of ∇fmust be same.

So,

b=a

Hence, it is proved that the rectangle must be a square in order to have its area maximum.

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