\

Step 1 :

(a)

Thu function .

Differentiate with respect to x:

Determination of critical points:

Since  is a polynomial it is continuous for all real numbers.

Thus, the critical points exist when .

Equate to zero:

The critical points are and the test intervals are .

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

Thus, The function is increasing on the intervals and .

And The function is decreasing on the interval and .

Step 2 :  

(b)

is changes its sign from negative to positive, hence f has a local minimum at .

Local minimum is .

is changes its sign from positive to negative, hence f has a local maximum at .

Local maximum is  .

is changes its sign from negative to positive, hence f has a local minimum at .

Local minimum is .

Step 3 :  

(c)

Differentiate with respect to x:

Determination of concavity and inflection points : 

Equate to zero:

Thus, the inflection points are split the intervals into and .

\ Interval \	Test Value	Sign of	Concavity
		\ \	Up
		\ \	Down
		\ \	\ Up \

Thus, the graph is concave up in the interval and .

The graph is concave down in the interval .

Inflection points :

Inflection points .

(a)

Increasing on the intervals and .

Decreasing on the interval and .

(b)

Local maximum is  .

Local minimum is .

(c)

Concave up on the interval and .

Concave down on the interval .

Inflection points .