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- Stationary definition is - fixed in a station, course, or mode: immobile. How to use stationary in a sentence. Stationary or stationery?
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Money 5 5 – personal accounting app. 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.[1][2][3] 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.
Turning points[edit]
A turning point is a point at which the derivative changes sign.[2] 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 x↦x3{displaystyle xmapsto x^{3}} has a stationary point at x = 0, which is also an inflection point, but is not a turning point.[3]
Classification[edit]
Isolated stationary points of a C1{displaystyle C^{1}} real valued function f:R→R{displaystyle fcolon mathbb {R} to mathbb {R} } 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.
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 are not local extremum—are known as saddle points.
By Fermat's theorem, global extrema must occur (for a C1{displaystyle C^{1}} function) on the boundary or at stationary points.
Curve sketching[edit]
Determining the position and nature of stationary points aids in curve sketching of differentiable functions. Solving the equation f'(x) = 0 returns the x-coordinates of all stationary points; the y-coordinates are trivially the function values at those x-coordinates.The specific nature of a stationary point at x can in some cases be determined by examining the second derivativef'(x): Sidify music converter 1 1 0 download free.
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- If f'(x) < 0, the stationary point at x is concave down; a maximal extremum.
- If f'(x) > 0, the stationary point at x is concave up; a minimal extremum.
- If f'(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 function f(x) = x3. There is a clear change of concavity about the point x = 0, and we can prove this by means of calculus. The second derivative of f is the everywhere-continuous 6x, and at x = 0, f′′ = 0, and the sign changes about this point. So x = 0 is a point of inflection.
More generally, the stationary points of a real valued function f:Rn→R{displaystyle fcolon mathbb {R} ^{n}to mathbb {R} } are thosepoints x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero.
Example[edit]
For the function f(x) = x4 we have f'(0) = 0 and f'(0) = 0. Even though f'(0) = 0, this point is not a point of inflection. The reason is that the sign of f'(x) changes from negative to positive.
For the function f(x) = sin(x) we have f'(0) ≠ 0 and f'(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 of f'(x) does not change; it stays positive.
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For the function f(x) = x3 we have f'(0) = 0 and f'(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 of f'(x) does not change; it stays positive.
See also[edit]
References[edit]
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- ^Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). New York: McGraw-Hill. p. 236. ISBN0-07-010813-7.
- ^ abSaddler, David; Shea, Julia; Ward, Derek (2011), '12 B Stationary Points and Turning Points', Cambridge 2 Unit Mathematics Year 11, Cambridge University Press, p. 318, ISBN9781107679573
- ^ ab'Turning points and stationary points'. TCS FREE high school mathematics 'How-to Library'. Retrieved 30 October 2011.
External links[edit]
- Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio at cut-the-knot
Stationary Definition
