$\"\"$

The function is $\"\"$ .

Find the intercepts :

To find the $\"\"$ -intercept, substitute $\"\"$ in the function.

$\"\"$

There is no real solution for $\"\"$ .

There is no $\"\"$ -intercept.

To find the $\"\"$ -intercept, substitute $\"\"$ in the function.

$\"\"$

$\"\"$ does not exist.

There is no $\"\"$ -intercept.

$\"\"$

Find the relative extrema for the function $\"\"$ :

Find the extrema for $\"\"$ .

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

$\"\"$

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

$\"\"$

$\"\"$ .

Find the critical numbers by equating $\"\"$ to $\"\"$ .

$\"\"$ .

$\"\"$

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

Critical numbers are $\"\"$ and $\"\"$ .

The relative extrema value is: $\"\"$ 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 we make $\"\"$ .

$\"\"$ .

No inflection point.

$\"\"$

Critical numbers are $\"\"$ and $\"\"$ .

Relative extrema points exist at critical numbers.

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

\ \

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

Test value	Sign of $\"\"$	Conclusion
$\"\"$	\ $\"\"$ \	Relative minimum
$\"\"$	\ $\"\"$ \	Relative maximum

The function has relative maximum at $\"\"$ .

Substitute $\"\"$ in $\"\"$ .

$\"\"$ .

Relative maximum point is $\"\"$ .

The function has relative minimum at $\"\"$ .

Substitute $\"\"$ in $\"\"$ .

$\"\"$ .

Relative minimum point is $\"\"$ .

$\"\"$

Find asymptote of function $\"\"$ .

To find horizontal asymptote, determine $\"\"$ .

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

As $\"\"$ , Then $\"\"$ .

$\"\"$

Thus, there is no horizontal asymptote.

To find vertical asymptote,equate denominator to zero.

$\"\"$

Imaginary roots are not consider.

There is no vertical asymptote.

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.

Observe the graph ,

The intercepts are $\"\"$ , $\"\"$ does not exists.

Relative maximum point is $\"\"$ .

Relative minimum point is $\"\"$ .

There is no possible inflection point.

No horizontal asymptote.

No vertical asymptote.

Slant asymptote is $\"\"$ .

$\"\"$

No intercepts.

Relative maximum point is $\"\"$ .

Relative minimum point is $\"\"$ .

There is no possible inflection point.

No horizontal asymptote.

No vertical asymptote.

Slant asymptote is $\"\"$ .