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For a differentiable function $f(x)$, how does the first derivative test determine a local extremum at critical point $x_0$ ($f'(x_0) = 0$)?

Study flashcard from the deck Gaokao essentials: original practice.

Answer

If $f'(x)$ changes from positive to negative at $x_0$, $f(x_0)$ is a local maximum; if from negative to positive, a local minimum.

Explanation

If $f'(x)$ does not change sign around $x_0$, $x_0$ is an inflection point or saddle rather than a local extremum.

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