$\"\"$

The function is $\"\"$ .

Intercept:

Find the $\"\"$ -intercept, $\"\"$ .

$\"\"$

$\"\"$ .

The $\"\"$ -intercept is $\"\"$

Find the $\"\"$ -intercept, $\"\"$ .

$\"\"$

The $\"\"$ -intercept is $\"\"$ .

$\"\"$

Find the extrema for $\"\"$ .

Differentiate on each side with respect to $\"\"$ .

$\"\"$

Quotient rule of derivatives : $\"\"$ .

$\"\"$

$\"\"$ .

$\"\"$

$\"\"$ .

To find the critical numbers, evaluate $\"\"$ .

$\"\"$ .

$\"\"$

$\"\"$ and $\"\"$ .

$\"\"$ , $\"\"$ and $\"\"$ .

The critcal point are $\"\"$ , $\"\"$ and $\"\"$ .

$\"\"$

To find the points of inflection of the graph $\"\"$ , then either $\"\"$ .

The first derivative of $\"\"$ is $\"\"$ .

Differentiate on each side with respect to $\"\"$ .

$\"\"$ .

$\"\"$

$\"\"$ .

The second derivative of $\"\"$ is $\"\"$ .

To find inflection points, evaluate $\"\"$ .

$\"\"$ .

$\"\"$

$\"\"$ , $\"\"$ and $\"\"$ .

Imaginary roots are not considered.

The possible inflection points occurs at $\"\"$ .

$\"\"$

The critcal points are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

Relative extrema points exist at critical numbers.

Substitute $\"\"$ in the function.

$\"\"$

$\"\"$ .

Substitute $\"\"$ in the function.

$\"\"$

$\"\"$ .

Substitute $\"\"$ in the function.

$\"\"$

$\"\"$ .

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

\ \

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Test value	Sign of $\"\"$	Conclusion
$\"\"$	\ $\"\"$ \	No conclusion
$\"\"$	\ $\"\"$ \	Relative minimum
$\"\"$	\ $\"\"$ \	Relative maximum

Relative maximum point is $\"\"$

Relative minimum point is $\"\"$ .

$\"\"$

Find asymptote of function $\"\"$ .

To find horizontal asymptote $\"\"$ .

$\"\"$

Take common terms numerator and denominator by $\"\"$ .

$\"\"$

Substitute As $\"\"$ Then $\"\"$ .

$\"\"$

No horizontal asymptote.

To find vertical asymptote,equate denominator to zero.

$\"\"$

The vertical asymptote is $\"\"$ and $\"\"$ .

Find the slant asymptote by long division method.

$\"\"$

Therefore, the function is reduced as $\"\"$ .

The slant asymptote is the polynomial part of the reduced expreession.

Therefore, slant asymptote is $\"\"$ .

$\"\"$

Graph :

Graph the function $\"\"$ .

$\"\"$

Note:The dashed lines indicates horizontal asymptote.

Intercepts is $\"\"$ .

The critcal points are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

Relative maximum point is $\"\"$

Relative minimum point is $\"\"$ .

The inflection points occurs at $\"\"$ .

No horizontal asymptote.The vertical asymptote is $\"\"$ and $\"\"$ .

$\"\"$

Intercepts is $\"\"$ .

The critcal points are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

Relative maximum point is $\"\"$

Relative minimum point is $\"\"$ .

The inflection points occurs at $\"\"$ .

No horizontal asymptote.The vertical asymptote is $\"\"$ and $\"\"$ .