Plot of y = x3 with inflection point of (0,0).

In differential calculus, an inflection point, or point of inflection (or inflexion) is a point on a curve at which the curvature changes sign. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry A negative number is a Number that is less than zero, such as −2 The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. In Mathematics, a concave function is the negative of a Convex function. If one imagines driving a vehicle along the curve, it is a point at which the steering-wheel is momentarily "straight", 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. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change
• a point (x,y) on a function, f(x), at which the first derivative, f'(x), is at an extremum, i. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that e. a minimum or maximum. (This is not the same as saying that y is at an extremum).
• a point on a curve at which the tangent crosses the curve itself. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation.
Plot of y = x3, rotated, with tangent line at inflection point of (0,0).

Note that since the first derivative is at an extremum, it follows that the second derivative, f''(x), is equal to zero, but the latter condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order non-zero derivative to be of odd order (third, fifth, etc. ). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection. (An example of such a function is y = x4).

It follows from the definition that the sign of f'(x) on either side of the point (x,y) must be the same. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection. A negative number is a Number that is less than zero, such as −2 A negative number is a Number that is less than zero, such as −2

Points of inflection can also be categorised according to whether f'(x) is zero or not zero.

• if f'(x) is not zero, the point is a non-stationary point of inflection
Plot of y = x4 - x with tangent line at non-inflection point of (0,0). In Mathematics, particularly in Calculus, a stationary point is an input to a function where the Derivative is zero (equivalently the In Mathematics, a saddle point is a point in the domain of a function of two variables which is a Stationary point but not a Local extremum

An example of a saddle point is the point (0,0) on the graph y=x³. The tangent is the x-axis, which cuts the graph at this point.

A non-stationary point of inflection can be visualised if the graph y=x³ is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar

Note that an inflection point is also called an ogee, although this term is sometimes applied to the entire curve which contains an inflection point. Ogee is a shape consisting of a concave arc flowing into a convex arc so forming an S-shaped curve with vertical ends