I believe what this question is saying (albeit poorly) is that you have 4 possible ways to encode what happens on a given interval between critical points, and not that the sequence you would submit as an answer consists of 4 digits.
For a basic example, let's consider [tex]f(x)=x^3[/tex]. We have
[tex]f'(x)=3x^2[/tex]
so that [tex]x=0[/tex] is the only critical point, and
[tex]f''(x)=6x[/tex]
so that [tex]x=0[/tex] is the only inflection point.
[tex]f'(x)>0[/tex] for all [tex]x[/tex], while [tex]f''(x)<0[/tex] for [tex]x<0[/tex], and vice versa.
We have two intervals to work with, [tex](-\infty,0)[/tex] and [tex](0,\infty)[/tex]. So in this case, the answer would be 2 over [tex](-\infty,0)[/tex] and 1 over [tex](0,\infty)[/tex].
That's a guess at any rate. Also, I don't know how helpful this is 4 days after the fact...
