In differential calculus, an inflection point, point of inflection, flex, or inflection (inflexion) is a point on a curve at which the curvature or concavity changes sign from plus to minus or from minus to plus. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a vehicle along a winding road, inflection is the point at which the steering-wheel is momentarily "straight" when being turned from left to right or vice versa.
The following are all equivalent to the above definition:
a point on a curve at which the second derivative changes sign. This is very similar to the previous definition, since the sign of the curvature is always the same as the sign of the second derivative, but note that the curvature is not the same as the second derivative.
a point (x, y) on a function, f(x), at which the first derivative, f′(x), is at an extremum, i.e. a (local) minimum or maximum. (This is not the same as saying that y is at an extremum).
a point p on a curve at which the tangent crosses the curve at that point. For an algebraic curve, this means a non singular point where the multiplicity of the intersection at p of the tangent line and the curve is odd and greater than 2.