The local extrema of the first derivative determining the inflection points makes sense; for $x = c$ to be a local extremum for the first derivative, the first derivative's derivative (i.e., the second derivative) would need to change signs as it passes through $x = c$ and $f''(c)=0$. [Wikipedia seems to agree](https://en.wikipedia.org/wiki/Inflection_point):

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> the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative, f′, has an isolated extremum at x
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The reason I ask is because of a multiple choices question from my highschool math book:

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> Which statement verifies that the function $f$ has a point of inflection at $x = c$?
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The two answers I was stuck on were

1. The derivative $f'$ has a local maximum at $x = c$
2. The sign of $f''$ changes at $x=c$

The answer is 2, but I don't really understand why 1 isn't correct. Maybe because inflection points can also happen if the first derivative has local minima? Though with the way the question is worded this shouldn't matter.