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I'm getting slightly confused with how critical points, inflection points, and relative extrema are all related to each other in a graph. Let me tell you what I think I know, and please correct me if I'm wrong.

Relative Extrema is the local maximum or minimum of a 'hill' of a graph of a polynomial of the second degree or higher. In other words, a relative extrema (extremum? extreme?) is a point where the derivative is 0. This can be found by finding where the derivative changes signs. Can this also be found just by looking where the derivative is 0? Is that the same thing? Which is easier?

Critical Points, also known as stationary points(?), is any point where the derivative is equal to 0. This can be found using the same method as above.

Inflection Points is the point where the rate of change of the derivative of the graph switches signs. In other words, the point where the curvature of the graph changes from concave up to down or vice versa. This can be found by finding the double derivative and finding when its sign changes/finding its zeroes (which is easier?).

Now my main problem is figuring out where these overlap. So these are my questions that I can't really get to the bottom of:

  • Are all relative extrema critical points?

  • Are all critical points relative extrema?

  • Can a critical point ever not be a relative extrema?

  • Will an inflection point always be distinct from a relative extrema and a critical point?

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  • All relative extrema are critical points.
  • However, not all critical points are relative extrema. For example plot $f(x)=x^3$ and note that $f'$ is zero at $x=0$, yet it is neither a relative maximum nor a relative minimum. In higher dimensions, saddle points are another example of critical points that are not relative extrema.
  • Consider $f(x)=x^5$. Its second derivative is $f''(x)=20x^3$, which changes sign at $x=0$. Its first derivative is $f'(x)=5x^4$ which is zero at $x=0$, so it is also a critical point.
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To answer my question about the difference between finding the zeroes and finding where a derivative changes signs, I've found from solving a few problems that finding the roots of a derivative isn't always sufficient, particularly when dealing with odd degree polynomials with stretches that lay nearly flat against the x-axis (I don't know how to describe this better). In a case like that, the derivative might have multiple zeroes, but only the roots farthest from zero might be the relative extrema.

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