Let the problem be stated as follows:

Suppose you buy 100 feet of wire fencing. What’s the largest rectangular area garden you can enclose with it?

Solution:

Let x be the width of the garden.

Let y be the height of the garden.

Then the perimeter of the rectangle must be equal to 100ft.

2x + 2y = 100

x + y = 50

y = 50 - x --------------------- > 1

Let the area be A and is given by

A = xy

using eqn(1)

A = x(50 - x)

A(x) = 50x - x²

is the model of our problem.

Now diff wrt x

For maxima/minima,

Let us check if the above x value leads to a maximum/minimum.

For that, let us find the second derivative,

diff wrt x

plugin the x value

PFA