The condition are :

.

.

.

.

.

.

From the conditions,

The value of the function at  is .

 represents that  is concave downward on .

 represents as  tends to  then  approaches to .

 represents as  tends to  then  approaches to .

 and  represents that  has a vertical asymptote at .

So  is not defined for .

Graph :

Graph the function such that all the conditions are satisfied :

Graph of the function satisfying all the conditions is

.