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

An inflection point, or point of inflection (or inflexion) can be defined in any of the following ways: Image File history File links I created this graph using software I wrote and grant full license to anyone to use, copy, and distribute. ... Image File history File links I created this graph using software I wrote and grant full license to anyone to use, copy, and distribute. ...

• a point on a curve at which the tangent crosses the curve itself

• a point on a curve at which the curvature changes sign. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. 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.

• 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 minimum or maximum. (This is not the same as saying that y is at an extremum, and in fact implies that y is not at an extremum).

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 = x^4 -x). A spatial point is an entity with a location in space but no extent (volume, area or length). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... Curvature is the amount by which a geometric object deviates from being flat. ... Sign can denote any of the following: Look up sign in Wiktionary, the free dictionary In astrology sign is often used to mean the Sun sign. ... In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ... Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, the derivative is one of the two central concepts of calculus. ... In mathematics, the derivative is one of the two central concepts of calculus. ... In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number). ... Image File history File links I created this graph using software I wrote and grant full license to anyone to use, copy, and distribute. ... Image File history File links I created this graph using software I wrote and grant full license to anyone to use, copy, and distribute. ... Zero can refer to several things. ...

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. In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary Negative and non-negative numbers Negative (photography) In optics, diverging lenses are also called negative lenses. ...

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

• if f'(x) is zero, the point is a stationary point of inflection, also known as a saddle-point

• 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).

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. Stationary pts (red pluses) and inflection pts (green circles). ... Plot of y = x3 with a saddle-point at (0,0). ... Image File history File links I created this graph using a program I wrote and grant everyone full access to use, publish, and distribute it. ... Image File history File links I created this graph using a program I wrote and grant everyone full access to use, publish, and distribute it. ...

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 the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

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 Arch Ogee is a shape consisting of a concave arc flowing into a convex arc, so forming an S-shaped curve with vertical ends. ...

See also

• Derivative
• Saddle point
• Stationary point