I was looking at a homework for a calc class I have online, couldn't find it, and searched online. I found this answer to these questions:
What is the minimum number of inflection points that must exist between (and not at ) two critical points of a non-constant differentiable polynomial?
A: The minimum number of inflection points is 0, because the critical points may not be of the form max-min.
What is the maximum number of inflection points that can exist between two critical points of a differentiable function?
A: Only one.
Both of these answers were incorrect. The justification for this was:
The correct answer is 1 because if you have two critical points that means there is either 2 maximums, 2 minimums or 1 maximum and 1 minimum. In any of these cases there has to be at least 1 inflection point.
and
- The correct answer is "There is no maximum" because there can be endless number of inflection points. The concavity can change a million times between two critical points.
In addition, the question:
Where does the function $R(x) = {(x^2-9)(x^2-6x+5)\over (x-3)(x^2-2x-3)(x-1)(x^2+4)}$ have an asymptote?
A. I got it wrong when answering x=1,3 instead of their answer of x=-1,3.
If you look at the graph it shows x=-1,1,3
Definition of Critical Point: Critical points are the places on a graph where the derivative equals zero or is undefined. Interesting things happen at critical points.
The question I have is: is the answer on StackExchange wrong, is the answer on the homework site wrong and how?