(a)

The completed table :

Height,	Length and Width,	Volume,

Observe the above table :

The maximum volume occurs at , i.e, .

(b)

From the table :

Volume of the box :

.

(c)

Volume .

Derivative on each side with respect to .

Find the crirical numbers by equating derivative to zero.

 and 

 and .

Thus, the critical numbers are  and .

.

Derivative on each side with respect to .

.

If , then .

By second derivative test,  is a maximum value.

Substitute  in .

.

Thus, the maximum volume is .

If , then .

By the second derivative test,  is a minimum value.

The volume function is .

Draw a coordinate plane.

Graph the function .

Graph :

Observe the graph :

The maximum volume occurs at  and the maximum volume is .

(a)

The completed table :

Height,	Length and Width,	Volume,

The maximum volume occurs at , i.e, .

(b)

Volume .

(c)

The critical numbers are  and .

The maximum volume is .

(d)

Graph of the function  :

The maximum volume occurs at  and the maximum volume is .