Inflection points

Definition and criteria for inflection points.

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Second derivative and convexity/concavity Characterization of convexity/concavity via the second derivative.
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Study Inflection points

Definition 1. \(f:(a,b)\to\mathbb{R}\) has inflection point at \(x_0\in(a,b)\), if the second derivative changes its sign.

Theorem 2. Criteria for inflection points:

  • If \(f\) is two times differentiable and has an inflection point at \(x_0\), then \(f''(x_0)=0\).

  • If \(f\) is three times differentiable, \(f''(x_0)=0\) and \(f'''(x_0)\neq0\), then \(f\) has an inflection point at \(x_0\).

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