Let
$$f ( x ) = \frac { x ^ { 4 } } { ( x - 1 ) ( x - 2 ) \cdots ( x - n ) }$$
where the denominator is a product of $n$ factors, $n$ being a positive integer. It is also given that the $X$-axis is a horizontal asymptote for the graph of $f$. Answer the independent questions below by choosing the correct option from the given ones.\\
a) How many vertical asymptotes does the graph of $f$ have?

Options: $n$ \quad less than $n$ \quad more than $n$ \quad impossible to decide

Answer: $\_\_\_\_$\\
b) What can you deduce about the value of $n$?

Options: $n < 4$ \quad $n = 4$ \quad $n > 4$ \quad impossible to decide

Answer: $\_\_\_\_$\\
c) As one travels along the graph of $f$ from left to right, at which of the following points is the sign of $f ( x )$ guaranteed to change from positive to negative?

Options: $x = 0$ \quad $x = 1$ \quad $x = n - 1$ \quad $x = n$

Answer: $\_\_\_\_$\\
d) How many inflection points does the graph of $f$ have in the region $x < 0$?

Options: none\quad 1\quad more than 1\quad impossible to decide

(Hint: Sketching is better than calculating.)

Answer: $\_\_\_\_$