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Inmathematics, particularly incalculus, astationary point of adifferentiable function of one variable is a point on thegraph of the function where the function'sderivative is zero.^[1][2][3] Informally, it is a point where the function "stops" increasing or decreasing (hence the name).

For a differentiablefunction of several real variables, a stationary point is a point on thesurface of the graph where all itspartial derivatives are zero (equivalently, thegradient has zeronorm).The notion of stationary points of areal-valued function is generalized ascritical points forcomplex-valued functions.

Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where thetangent is horizontal (i.e.,parallel to thex-axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to thexy plane.

The notion of astationary point allows the mathematical description of anastronomical phenomenon that was unexplained before the time ofCopernicus. A stationary point is the point in the apparent trajectory of the planet on thecelestial sphere, where the motion of the planet seems to stop, before restarting in the other direction (seeapparent retrograde motion). This occurs because of the projection of the planetorbit into theecliptic circle.

Aturning point of adifferentiable function is a point at which the derivative has anisolated zero and changes sign at the point.^[2] A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). A turning point is thus a stationary point, but not all stationary points are turning points. If the function is twice differentiable, the isolated stationary points that are not turning points are horizontalinflection points. For example, the function${\displaystyle x\mapsto x^3}$ has a stationary point atx = 0, which is also an inflection point, but is not a turning point.^[3]

Isolated stationary points of a${\displaystyle C^1}$ real valued function${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ are classified into four kinds, by thefirst derivative test:

• alocal minimum (minimal turning point orrelative minimum) is one where the derivative of the function changes from negative to positive;
• alocal maximum (maximal turning point orrelative maximum) is one where the derivative of the function changes from positive to negative;

• arisingpoint of inflection (orinflexion) is one where the derivative of the function is positive on both sides of the stationary point; such a point marks a change inconcavity;
• afalling point of inflection (orinflexion) 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.

The first two options are collectively known as "local extrema". Similarly a point that is either a global (or absolute) maximum or a global (or absolute) minimum is called a global (or absolute) extremum. The last two options—stationary points that arenot local extrema—are known assaddle points.

ByFermat's theorem, global extrema must occur (for a${\displaystyle C^1}$ function) on the boundary or at stationary points.

Determining the position and nature of stationary points aids incurve sketching of differentiable functions. Solving the equationf′(x) = 0 returns thex-coordinates of all stationary points; they-coordinates are trivially the function values at thosex-coordinates.The specific nature of a stationary point atx can in some cases be determined by examining thesecond derivativef″(x):

• Iff″(x) < 0, the stationary point atx is concave down; a maximal extremum.
• Iff″(x) > 0, the stationary point atx is concave up; a minimal extremum.
• Iff″(x) = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point.

A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points (if the function is defined and continuous between them).

A simple example of a point of inflection is the functionf(x) =x³. There is a clear change of concavity about the pointx = 0, and we can prove this by means ofcalculus. The second derivative off is the everywhere-continuous 6x, and atx = 0,f″ = 0, and the sign changes about this point. Sox = 0 is a point of inflection.

More generally, the stationary points of a real valued function${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$ are thosepointsx₀ where the derivative in every direction equals zero, or equivalently, thegradient is zero.

For the functionf(x) =x⁴ we havef′(0) = 0 andf″(0) = 0. Even thoughf″(0) = 0, this point is not a point of inflection. The reason is that the sign off′(x) changes from negative to positive.

For the functionf(x) = sin(x) we havef′(0) ≠ 0 andf″(0) = 0. But this is not a stationary point, rather it is a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign off′(x) does not change; it stays positive.

For the functionf(x) =x³ we havef′(0) = 0 andf″(0) = 0. This is both a stationary point and a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign off′(x) does not change; it stays positive.

For the functionf(x) = 0, one hasf′(0) = 0 andf″(0) = 0. The point 0 is a non-isolated stationary point which is not a turning point nor a horizontal point of inflection as the signs off′(x) andf″(x) do not change.

The functionf(x) =x⁵ sin(1/x) forx ≠ 0, andf(0) = 0, gives an example wheref′(x) andf″(x) are both continuous,f′(0) = 0 andf″(0) = 0, and yetf(x) does not have a local maximum, a local minimum, nor a point of inflection at 0. So,0 is a stationary point that is not isolated.

• Optimization (mathematics)
• Fermat's theorem
• Derivative test
• Fixed point (mathematics)
• Saddle point

• Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio atcut-the-knot

• Differential calculus

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