(a)

The function .

Find the - intercept :

Consider .

Find the - intercept, by substituting in .

.

The - intercept is .

(b)

Find for what numbers the graph of is decreasing in the interval is .

The function .

Apply derivative on each side with respect to  .

Determination of critical points :

The critical points exist when .

Equate to zero:

Solve in the interval .

General solution of is , where is an integer.

General solution is

If , .

If , .

If ,

The solutions are , and in the interval .

The critical points are , and and the test intervals are .

Interval	Test Value	Sign of	Conclusion
		\ \	Increasing
		\ \	Decreasing

The function is Decreasing over the interval .

(c)

The function and the interval is .

The critical points are and and .

Find the values of at these critical points.

.

.

.

Compare the values of to find the absolute minimum.

Absolute minimum value is .

(d)

The function and the interval is .

General solution of is , where is an integer.

General solution : .

If , .

If , .

If , .

The solutions are  and in the interval .

(e)

The function and the interval is .

General solution of  is , where is an integer.

General solution : .

If , .

If , .

If , .

The solutions are , and in the interval .

General solution of is , where is an integer.

General solution : .

If , .

If , .

The solutions are and in the interval .

(f)

General solution of is , where is an integer.

General solution : .

If , .

If , .

If , .

The solutions are   in the interval .

(g)

 Find the - intercept :

The function .

Find the - intercept, by substituting in .

General solution of  is , where is an integer.

The - intercepts are , where is an integer.

(a). The - intercept is .

(b). The function is Decreasing over the interval .

(c). Absolute minimum value is .

(d). The solutions are  and in the interval .

(e). For ,the solutions are , and in the interval

For , the solutions are and in the interval .

(f). The solutions are   in the interval ..

(g).The - intercepts are , where is an integer.