$\"\"$

The function is $\"\"$ .

Find the horizontal asymptote.

$\"\"$

Therfore the horizontal asymptote is $\"\"$ .

Find the vertical asymptote.

To find the vertical asymptote, equate denominator of the function to zero.

So $\"\"$ .

Here the roots are imaginary.

Therefore there is no vertical asymptote.

$\"\"$

The function is $\"\"$ .

$\"\"$

Find the critical points.

Thus, the critical points exist when $\"\"$ .

$\"\"$

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

The test intervals are $\"\"$ , $\"\"$ and $\"\"$ .

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

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

The function is increasing on the interval $\"\"$ .

The function is decreasing on the intervals $\"\"$ and $\"\"$ .

Find the local maximum and local minimum.

The function $\"\"$ has a local minimum at $\"\"$ , because $\"\"$ changes its sign from negative to positive.

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

$\"\"$

Local minimum is $\"\"$ .

The function $\"\"$ has a local maximum at $\"\"$ , because $\"\"$ changes its sign from positive to negative.

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

$\"\"$

Local maximum is $\"\"$ .

$\"\"$

$\"\"$ .

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

$\"\"$ .

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

The test intervals are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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

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

The graph is concave up on the intervals $\"\"$ and $\"\"$ .

The graph is concave down on the intervals $\"\"$ and $\"\"$ .

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

$\"\"$

Graph :

Graph the function $\"\"$ :

$\"\"$

$\"\"$

Horizontal asymptote $\"\"$ .

The function is increasing on $\"\"$ and decreasing on $\"\"$ and $\"\"$ .

The graph is concave up on $\"\"$ and $\"\"$ and concave down on $\"\"$ and $\"\"$ .

Graph of the function $\"\"$ is

$\"\"$ .