Step 1:

The total material is required to fence a rectangular field is ft.

Let the rectangular field has length ft and breadth ft.

The perimeter of the rectangular field is .

.

The area of the rectangular field is .

Substitute in .

.

The area of the rectangular field is always positive.

and .

is  positive on the interval .

Step 2:

Find the dimensions of the rectangular field, that will enclose the maximum area.

Apply derivative on each side with respect to .

.

To find the critical numbers by equating .

.

Step 3:

The maximum value of occurs at either at critical number or at end point of the interval .

Substitute in .

.

Substitute in .

.

Substitute in .

.

The area maximum at length of rectangular field is ft.

Substitute in .

ft.

The dimensions of the rectangular field is ft and ft.

Solution:

The dimensions of the rectangular field is ft and ft.

(2)

Step 1:

The rectangular field area is square ft.

Let the rectangular field has length ft and breadth ft.

The area of the rectangular field is .

.

The perimeter of the rectangular field is .

Substitute in .

.

The perimeter of the rectangular field is always positive.

.

Perimeter is  positive on the interval .

Step 2:

Find the dimensions of the rectangular field, that will require least amount of fencing.

Apply derivative on each side with respect to .

.

To find the critical numbers by equating .

.

Step 3:

The minimum value of occurs at either at critical number or at end point of the interval .

is negative for .

for .

Since is decreasing for all to the left of the critical number and increasing for all to the right, must give rise to an absolute minimum.

Substitute in .

. \ \

The dimensions of the rectangular field are ft and ft.

Solution:

The dimensions of the rectangular field are ft and ft.

Substitute in .

ft.

The dimensions of the rectangular field is ft and ft.

Solution:

The dimensions of the rectangular field is ft and ft.

Therefore .

.

Given product of one number and square of other number is maximum.

So,Product  .

.

Now we have to find the derivative of the obtained function.

.

is not possible because .

So .

Therefore the two non negative numbers are and .

Step 2:

Given product of one number and square of other number is maximum.

The product of these two numbers is 108.

Solution:

Therefore the two non negative numbers are and .

The product  is 108.

4)

Step 1: Given function . Given Points .

Plot the graph of given function .

Graph:

The graph of .

.

The points closest to the point are .

Solution:

The points closest to the point are .