An experiment with a new type of problems.

Algebra Level 5

Let \(f(x) =\) \( \dfrac { { x }^{ 4 } }{ (x-1)(x-2)...(x-n) }\) where the denominator is a product of n factors, n being a positive integer.

It is also a given that the X-axis is a horizontal asymptote for the graph of f. The following questions are independent of one another.

a)How many vertical asymptotes does the graph of f have?

Options:

1) n

2) less than n

3) more than n

4) impossible to decide

Answer to this is denoted by a, i.e, if you think 4 is the right answer, then a=4.

b) What can you deduce about the value of n?

Options:

1) n<4

2) n=4

3) n>4

4) impossible to decide

Answer is to this question is denoted by b.

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 (not necessarily continuously)?

Options:

1) x=0

2) x=1

3) x=n−1

4) x=n

Answer to this is denoted by c.

d) How many inflection points does the graph of f have in the region x<0?

Options:

1) none

2) more than 1

3) 1

4) impossible to decide.

Answer to this is denoted by d.

Find\(\begin{vmatrix} a & \quad c \\ b & \quad d \end{vmatrix}\)

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