The function is $\"\"$ .

Find the horizontal asymptote.

$\"\"$

Therefore the horizontal asymptotes are at $\"\"$ and $\"\"$ .

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 $\"\"$ .

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

$\"\"$

Find the critical points.

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

$\"\"$

Here the roots are imaginary, so there is no critical points.

But $\"\"$ , hence the function is increases.

Concavity :

$\"\"$ .

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

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

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

$\"\"$

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

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

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

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

The inflection point is $\"\"$

Graph :

Graph the function $\"\"$ :

$\"\"$

The horizontal asymptotes are $\"\"$ and $\"\"$ .

The function is increasing.

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

Graph of the function $\"\"$ is

$\"\"$ .