The function is .

(A)

Domain :

The function is .

All possible values of is the domain of the function.

The function is a polynomial function hence it is continuous for all the points.

Therefore the domain of the function is the set of all real numbers.

The domain of the function is .

(B)

Intercepts :

To find the -intercepts, substitute in the function.

.

Therefore the -intercept is .

To find the -intercepts, substitute in the function.

and .

Therefore the -intercepts are and .

(C)

Symmetry :

Substitute in the function.

Here .

Therefore the function is an even.

The function is symmetry about -axis.

(D)

Asymptotes :

There is no vertical asymptotes, since the function is continuous at all real numbers.

Horizontal asymptote :

There is no horizontal asymptote.

(E)

Intervals of increase or decrease :

The function is .

Differentiate on each side with respect to .

Find the critical points.

Since is a polynomial it is continuous at all the point.

Thus, the critical points exist when .

Equate  to zero.

and

, and .

The critical points are ,  and .

The test intervals are , , and .

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

The function is increasing on the intervals and .

The function is decreasing on the interval and .

(F)

Local Maximum and Minimum values :

The function has a local maximum at , because changes its sign from negative to positive.

Substitute in .

.

Local maximum is .

(G)

Concavity and point of inflection :

.

Differentiate on each side with respect to .

Find the inflection points.

Equate to zero.

and

and

and

The inflection points are and .

The test intervals are , , ,   and .

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

The graph is concave up in the intervals and .

The graph is concave down in the interval , and .

(H)

Graph :

Graph of the function  :

(A) Domain of the function is .

(B) -intercept is and -intercepts are and .

(C) No symmetry.

(D) No asymptotes.

(E)

The function is increasing on the intervals and .

The function is decreasing on the interval and .

(F) 

Local maximum is .

(G)

The graph is concave up in the intervals and .

The graph is concave down in the interval , and .

(H) Graph of the function is

.