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Chapter 3: Q. 63 (page 291)

Prove that the rectangle with the largest possible area given a fixed perimeter P is always a square.

Short Answer

Expert verified

The rectangle with the largest possible area given a fixed perimeter P is always a square has been proved.

Step by step solution

01

Step 1. Given information.

We have to prove that the rectangle with the largest possible area given a fixed perimeter P is always a square.

02

Step 2. Find the area.

Let x and y be the sides of the rectangle.

Thus,

P=2x+2y

and,

y=12(P鈭�2x)

The area is :

A=xy=x12(P鈭�2x)

03

Step 3. Derivative of the area.

Derivative of the area is:

A=x12(P鈭�2x)A鈥�=12P鈭�2x=0x=P4

Critical point is x=P4

Since,

A鈥�P4鈭�1>0andA鈥�P4+1<0

So, the first derivative gives us the local maximum at x=P4

Global maximum =0,P2

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Most popular questions from this chapter

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

Find the possibility graph of its derivative f'.

Sketch the graph of a function f with the following properties:

f is continuous and defined on R;

f has critical points at x = 鈭�3, 0, and 5;

f has inflection points at x = 鈭�3, 鈭�1, and 2.

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

role="math" localid="1648368106886" f(x)=sin(蟺2x)

Use the second-derivative test to determine the local extrema of each function fin Exercises 29-40. If the second-derivative test fails, you may use the first-derivative test. Then verify your algebraic answers with graphs from a calculator or graphing utility. (Note: These are the same functions that you examined with the first-derivative test in Exercises 39-50of Section 3.2.)

f(x)=(x-2)2(1+x)
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