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Inflection Point

Definition

Let ff be defined on an interval, and let x0x_0 be an interior point of it. The graph point (x0,f(x0))(x_0,f(x_0)) is an inflection point if the graph changes from convex to concave, or from concave to convex, at x0.x_0\text{.}

Interpretation

An inflection point marks a change in the curvature of a function: on one side of it the graph is convex, on the other side concave. It is a statement about the shape on both sides, not about a single value of a derivative.

The cubic x cubed over 3 minus x with its inflection point at 0, where it changes from concave on the left to convex on the right; the tangent at 0 crosses the curve.

For f(x)=13x3−xf(x)=\tfrac{1}{3}x^3-x, the second derivative f′′(x)=2xf''(x)=2x is negative to the left of 00 and positive to the right, so the graph changes from concave to convex at the inflection point (0,0)(0,0). The tangent there, y=−xy=-x, crosses the graph: on the concave side the graph lies below the tangent, on the convex side above it.

The point lies on the graph of the function, so its coordinates are (x0,f(x0))(x_0,f(x_0)). For a differentiable ff, convex means that f′f' is increasing and concave that f′f' is decreasing, so at an inflection point f′f' switches from decreasing to increasing or back: x0x_0 is a local extremum of f′f'. In the figure, f′(x)=x2−1f'(x)=x^2-1 has its minimum at 00.

Two slips go in opposite directions. The equation f′′(x0)=0f''(x_0)=0 is not enough: x4x^4 has f′′(0)=0f''(0)=0 but is convex on both sides of 00. And an inflection point need not satisfy f′′(x0)=0f''(x_0)=0, because f′′(x0)f''(x_0) need not exist: x∣x∣x|x| is concave on (−∞,0](-\infty,0] and convex on [0,∞)[0,\infty), but has no second derivative at 00. So the candidates are the zeros of f′′f'' together with the points where f′′f'' does not exist, and each candidate must be checked for a change of sign of f′′f''.

Key Results

  • If ff is twice differentiable at an inflection point x0,x_0\text{,} then f′′(x0)=0.f''(x_0)=0\text{.}
  • The equation f′′(x0)=0f''(x_0)=0 alone is not sufficient: the convexity must actually change at x0.x_0\text{.}

Examples

  • The function f(x)=x3f(x)=x^3 has an inflection point at x=0.x=0\text{.}
  • The function f(x)=x4f(x)=x^4 has no inflection point at x=0.x=0\text{.}

Prerequisites

Sources

Aliases: point of inflection, inflexion point Symbols: infinity

Last updated: 20 October 2026 03:02