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Theres these two related questions that are just mind boggling and I have no idea how to answer them...

  1. What is the minimum number of inflection points that must exist between (and not at ) two critical points of a non-constant differentiable polynomial?

  2. What is the maximum number of inflection points that can exist between two critical points of a differentiable function?

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We have to stress that critical points are not counted. Also, note than a critical point is an inflection point iff it is in between one maximum point and one minimum point. Than we have:

  1. The minimum number of inflection points is 0, because the critical points may not be of the form max-min.
  2. Only one.
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  • $\begingroup$ note that the answer choices for the second question are A. 1 B. "The same number as the degree of the function" C. 2 D. There is no maximum. Couldn't the answer also be B? $\endgroup$ – Panthy Jul 15 '14 at 23:55
  • $\begingroup$ No! If we can consider that alternative, we see that: $P_3(x)$ has one inflection point; $P_4(x)$ has two inflection point; ... So there cannot be as many inflection points as their degree. $\endgroup$ – Emin Jul 16 '14 at 0:07
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I'm not altogether clear on the question. First, I assume you are asking about inflection points between two adjacent critical points - otherwise a polynomial can have as many critical points as you like, and as many inflection points as you like between the first and the last critical point. Second (I assume this is what you mean but I think it is worth emphasizing), horizontal inflection points are critical points, so I assume you are basically interested in non-horizontal inflection points.

Having done that, critical points occur when $f'(x)=0$, and non-horizontal inflection points occur when $f'(x)\ne0$ and $f''(x)=0$.

Applying Rolle's Theorem to $f'$: if $f'(a)=f'(b)=0$, then there exists $c$ in $(a,b)$ such that $f''(c)=0$. So there is at least one inflection point between any two critical points.

Next, suppose that $f$ has degree $n$. Then $f''$ has degree $n-2$, so it has up to $n-2$ real zeros and hence up to $n-2$ inflection points. It is possible for all of these to lie between adjacent critical points of $f$. So there is no overall maximum: there can be as many inflection points as you like between two adjacent critical points. If you want an answer in terms of the degree of the polynomial, the maximum is the degree minus $2$.

An example with $n=5$: take $f(x)=3x^5-30x^4+110x^3-180x^2$. Then you can easily check that $f'(x)=0$ for $x=0$ and $x=4$ only; and $f''(x)=0$ for $x=1,2,3$.

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