$\"\"$

\

The condition are :

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

$\"\"$ for all $\"\"$ .

\

From the conditions,

\

The critical point is $\"\"$ .

\

$\"\"$ represents that $\"\"$ is increasing on $\"\"$ .

\

$\"\"$ represents that $\"\"$ is decreasing on $\"\"$ .

\

At $\"\"$ , $\"\"$ changes from negative to positive, so $\"\"$ has a local minimum at $\"\"$ .

\

$\"\"$ represents that $\"\"$ is concave upward on $\"\"$ .

\

$\"\"$ represents that $\"\"$ is concave downward on $\"\"$ .

\

$\"\"$ represents, the inflection point is $\"\"$ .

\

$\"\"$ represents horizontal asymptote at $\"\"$ .

\

$\"\"$ represents that $\"\"$ is an even function.

\

Graph :

\

Graph the function such that all the conditions are satisfied :

\

$\"\"$

\

$\"\"$

\

Graph of the function satisfying all the conditions is

\

$\"\"$ .