Inflection Point


An inflection point is a point on a curve at which the sign of the curvature (i.e., the concavity) changes. Inflection points may be stationary points, but are not local maxima or local minima. For example, for the curve y=x^3 plotted above, the point x=0 is an inflection point.

The first derivative test can sometimes distinguish inflection points from extrema for differentiable functions f(x).

The second derivative test is also useful. A necessary condition for x to be an inflection point is f^('')(x)=0. A sufficient condition requires f^('')(x+epsilon) and f^('')(x-epsilon) to have opposite signs in the neighborhood of x (Bronshtein and Semendyayev 2004, p. 231).

See also

Biflecnode, Curvature, Differentiable, Extremum, First Derivative Test, Local Maximum, Local Minimum, Second Derivative Test, Stationary Point Explore this topic in the MathWorld classroom

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Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, 2004.

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Inflection Point

Cite this as:

Weisstein, Eric W. "Inflection Point." From MathWorld--A Wolfram Web Resource.

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