Inflection Point
An inflection point is where a curve changes from concave up to concave down (or vice versa), marking a shift in bending direction.
Formula
f''(c) = 0 \text{ and concavity changes}
Definition
An inflection point is where a curve switches its bending direction, going from curving upward (like a bowl) to curving downward (like a dome), or vice versa; it is not necessarily a peak or valley. Formally, $f$ has an inflection point at $c$ if $f''(c) = 0$ (or is undefined) and $f''$ changes sign there, so the concavity of $f$ switches; $f''(c) = 0$ alone is not sufficient, the sign must actually change. In higher dimensions, inflection curves (not points) exist where the Gaussian curvature changes sign.
Example
An S-shaped road has an inflection point in the middle where it stops bending left and starts bending right. For $f(x) = x^3$: $f''(x) = 6x = 0$ at $x = 0$, and since $f'' < 0$ for $x<0$ and $f'' > 0$ for $x>0$, $x=0$ is an inflection point; but be careful, $x^4$ has $f''(0)=0$ with no inflection there because $f''$ never changes sign. The logistic function $f(x) = 1/(1 + e^{-x})$ has an inflection at $x = 0$, the point of maximum growth rate in logistic (S-curve) population models.
Key Insight
An inflection point is where the curve changes its "mood" from cupping up to cupping down, not where it peaks or bottoms out. Inflection points of a cumulative distribution function correspond to the mode of the underlying probability density, making them important in statistics and data analysis.