Step 1:

The rational function

Find the horizontal asymptote :

Since degree of numerator less than the degree of denominator, the horizontal asymptote .

Find the vertical asymptotes by solving zeros of denominator.

There is no vertical asymptotes.

Step 2:

Find the critical numbers by equate the first derivative to zero.

The function does not exist when .

The denominator does not have real roots.

Equate the numerator of to zero.

Substitute the values of in original function.

Critical points are and .

Consider the test intervals as  and

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

 The function is increasing on the interval .

And the function is decreasing on the intervals  and

Step 3:

Find the inflection points by equate the second derivative to zero.

The function does not exist when .

The denominator does not have real roots.

Equate the numerator of to zero.

and

and

Substitute the values of in original function.

Inflection points are , ,

Consider the test intervals as   and .

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

Graph:

Draw the coordinate plane.

Plot the critical points and inflection points of the curve.

Solution:

Horizontal asymptote .

.