The function is $\"\"$ .

Find the horizontal asymptote.

$\"\"$

Therefore the horizontal asymptote is at $\"\"$ .

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

So $\"\"$

$\"\"$

Here the roots of the function is imaginary, so there is no vertical asymptote.

The function is $\"\"$ .

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

$\"\"$

Find the critical points.

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

$\"\"$

The critical point is $\"\"$ .

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

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

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

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 $\"\"$ .

$\"\"$ .

Again apply derivative on each side with respect to $\"\"$ .

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

The inflection point is at $\"\"$ and $\"\"$ .

$\"\"$

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

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The graph is concave up on the interval $\"\"$ .

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

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

Graph :

Graph the function $\"\"$ :

$\"\"$

The horizontal asymptote is at $\"\"$ .

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

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

Graph of the function $\"\"$ is

$\"\"$ .