$\"\"$

The function is $\"\"$ .

Intercepts:

To find the $\"\"$ -intercept, $\"\"$ .

$\"\"$

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

Imaginary roots are not considered, so there is no $\"\"$ -intercept.

To 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 $\"\"$ .

The critical numbers are $\"\"$ and $\"\"$ .

$\"\"$

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

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

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

$\"\"$ .

$\"\"$

$\"\"$ .

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

To find inflection points we make $\"\"$ .

$\"\"$ .

Thus, there is no inflection points.

$\"\"$

The 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 maximum
$\"\"$	\ $\"\"$ \	Relative minimum

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

Substitute $\"\"$ in the function.

$\"\"$

Relative maximum point is $\"\"$ .

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

Substitute $\"\"$ in the function.

$\"\"$

Relative minimum point is $\"\"$ .

$\"\"$

Find asymptote of function $\"\"$ .

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

$\"\"$

No horizontal asymptote.

To find vertical asymptote,equate denominator to zero.

$\"\"$

The vertical asymptote is $\"\"$ .

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.

No $\"\"$ -intercept.

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

The critical numbers are $\"\"$ and $\"\"$ .

No inflection points.

Relative maximum point is $\"\"$ .

Relative minimum point is $\"\"$ .

No horizontal asymptote.

The vertical asymptote is $\"\"$ .

Slant asymptote is $\"\"$ .

$\"\"$

No $\"\"$ -intercept.

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

The critical numbers are $\"\"$ and $\"\"$ .

No inflection points.

Relative maximum point is $\"\"$ .

Relative minimum point is $\"\"$ .

No horizontal asymptote.

The vertical asymptote is $\"\"$ .

Slant asymptote is $\"\"$ .