The function $\"\"$ .

Find the horizontal asymptote.

$\"\"$

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

Find the vertical asymptote.

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

So $\"\"$ .

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

The function is $\"\"$ .

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

$\"\"$

Find the critical points.

Here $\"\"$ is never zero, but it is undefined at $\"\"$ .

The critical point is $\"\"$ .

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

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

So there is neither local maximum nor local minimum, since the function is decreasing on $\"\"$ .

Concavity :

$\"\"$ .

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

$\"\"$

$\"\"$ is never zero, but it is undefined at $\"\"$ .

There is no inflection points.

Consider the test interval are $\"\"$ and $\"\"$ .

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

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

Graph :

Graph the function $\"\"$ :

$\"\"$

The horizontal asymptote is $\"\"$ .

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

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

Graph of the function $\"\"$ is

$\"\"$ .