show that area of rectangle with fixed perimeter is greatest when rectangle is square


I have a question which asks you to show that, if the perimeter of a rectangle is fixed in length, the area of the rectangle is greatest when it is square. There is no answer to this in my book so I was hoping you could see if I have done it right....

A rectangle has 4 sides, 2 of length a and 2 of length b
The perimeter, P = 2a + 2b
a = P/2 - b
b = P/2 - a

The area of a rectangle, A, is a*b. Substituting b for P/2 -a:
A = a(P/2 - a)
A = Pa/2 - a^2

dA/da = P/2 - 2a

This function has a stationary point (maximum) when Da/da = 0
P/2 - 2a = 0
a = P/4

In other words the function for the area has a maximum when a = P/4, or when the length of the side is a quarter of the perimeter, or the rectangle has 4 sides of equal length (a square)

The second derivative of dA/da = P/2 - 2a
d^2A/da^2 = -2. If this is negative then the stationary point is going from positive to negative so it's a maximum point (and not a minimum or point of inflection)

Who is Participating?
I wear a lot of hats...

"The solutions and answers provided on Experts Exchange have been extremely helpful to me over the last few years. I wear a lot of hats - Developer, Database Administrator, Help Desk, etc., so I know a lot of things but not a lot about one thing. Experts Exchange gives me answers from people who do know a lot about one thing, in a easy to use platform." -Todd S.


And you can plot this:  A = Pa/2 - a^2     for P = 4 or 8

Experts Exchange Solution brought to you by

Your issues matter to us.

Facing a tech roadblock? Get the help and guidance you need from experienced professionals who care. Ask your question anytime, anywhere, with no hassle.

Start your 7-day free trial
that is the right approach
andiejeAuthor Commented:
many thanks
It's more than this solution.Get answers and train to solve all your tech problems - anytime, anywhere.Try it for free Edge Out The Competitionfor your dream job with proven skills and certifications.Get started today Stand Outas the employee with proven skills.Start learning today for free Move Your Career Forwardwith certification training in the latest technologies.Start your trial today
Math / Science

From novice to tech pro — start learning today.