An experiment with a new type of problems.
Let 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
The answer is -7.
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1 solution
Caesar said, "Apes alone weak. But Apes together, strong." The growth in overall toughness in a group of moderately easy problems can be seen here. But that's enough with the type of problem. Let's move on to the solution.
a = 1 because the function has n asymptotes at all natural numbers up to n.
b = 3 because the x axis acts as an asymptote when the degree of the denominator is greater than that of the numerator.
c = 3 because beyond x=n, the function is always positive. So, the function is negative between n-1 and n; and positive between n-2 and n-1. It's guaranteed to change from positive to negative as you move from left to write at x=n-1.
d = 2 because:
Let n be even. In that case, due to the x 4 part, you have 4 multiplicities at x=0 and you can verify with a rough graph that the function is concave up in this region. As x → − ∞ the function has to decay to zero and it has to approach 0 from above, so it must increase first. So, it becomes concave down and then eventually, it becomes concave up as x approaches infinity.
If n is odd, it approaches 0 − as x → − ∞ . As one traces the graph leftward from the origin, the function is initially as well as eventually concave down and must be concave up at least once in-between so as to approach the X-axis from below.