The function is and interval is .

Find the intercepts :

To find the - intercept, substitute in the function.

Solve  in the interval .

Let .

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Solution of the quadratic equation are

and .

and .

There are no solution for .

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General solution of is .

For , .

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  is not in the interval .

One solution is .

For , .

  is not in the interval .

Another solution is .

The solutions of  are and in the interval .

Therefore, the - intercepts are and .

To find the -intercept, substitute in the function.

The - intercept is .

Find the relative extrema for the function :

Consider .

Differentiate on each side with respect to .

To find the critical number, evaluate .

 and 

and

The solutions of are , and in the interval .

The solutions of are and in the interval .

Critical points occur at ,  and .

Find the points of inflection :

Consider .

Derivative on each side with respect to .

To find inflection points, equate to zero.

and .

The solutions of are and in the interval .

The solutions of are and in the interval .

If , then .

If , then .

If , then .

If , then .

The inflection points are , , and .

Find the asymptotes :

The function is .

Vertical asymptote :

The line is a vertical asymptote if .

Therefore, there is no vertical asymptote for the function .

Horizontal asymptote :

The line is a horizontal asymptote if .

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Limit does not exist.

Therefore, there is no horizontal asymptote for the function .

Find intervals of increase or decrease :

Critical points are ,  and .

Test interval are and .

Interval	Test value	Sign of	Conclusion
	\ \	\ \	Decrease
	\ \	\ \	Increase

The function is increasing in the interval .

The function is decreasing in the interval .

Perform second derivative test to identify the nature of the extrema.

Test value	Sign of	Conclusion
	\ \	No Conclusion
	\ \	Relative minimum
	\ \	No Conclusion

Fom the table of increase and decrease, the starts decreasing from a maximum point and increases upto a maximum point.

Hence Relative maximum occurs at and .

Relative minimum occurs at .

Find the value of the function at critical points.

If , .

If , .

If , .

Relative maximum at and .

Relative maximum at .

Find the Concavity of the function.

Inflection points occur at , , and .

Consider the test intervals as , and .

Interval	Test value	Sign of	Conclusion
		\ \	Concave Down
		\ \	Concave Up
		\ \	Concave Down

The graph is concave up on the intervals

The graph is concave down on the intervals and .

Using all the above characteristics of the function, graph the function in the interval .

Graph :

Draw a coordinate plane.

Graph the function in the interval .

Graph of the function in the interval is

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