Stationary point

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero). Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., parallel to the x-axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the xy plane. A turning point is a point at which the derivative changes sign. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. If the function is twice differentiable, the stationary points that are not turning points are horizontal inflection points. For example, the function has a stationary point at x = 0, which is also an inflection point, but is not a turning point. maxima and minima Isolated stationary points of a real valued function are classified into four kinds, by the first derivative test: a local minimum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive; a local maximum (maximal turning point or relative maximum) is one where the derivative of the function changes from positive to negative; a rising point of inflection (or inflexion) is one where the derivative of the function is positive on both sides of the stationary point; such a point marks a change in concavity; a falling point of inflection (or inflexion) is one where the derivative of the function is negative on both sides of the stationary point; such a point marks a change in concavity.

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Stationary point

Derivative test

In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain.

Inflection point

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

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