Stationary point
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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 (in 2D) or the plane tangent to the surface is parallel to the XY plane (in 3D). An equivalent definition is where the derivative of the function equals zero (known as a critical number).
An inflection point is a point where the concavity changes. A point of inflection is not necessarily a stationary point. All inflection points have the property of f''(x) = 0 but the reverse is not necessarily true.
Stationary points of a real valued function f: R → R are classified into four kinds:
- a minimal extremum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive;
- a maximal extremum (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
Determining the position and nature of stationary points aids in curve sketching, especially for continuous 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 derivative f''(x):
- If f''(x) < 0, the stationary point at x is a maximal extremum.
- If f''(x) > 0, the stationary point at x is 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 provided the function values exist around that point.
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 are those points x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero.
Example
At x1 we have f' (x) = 0 and f''(x) = 0. Even though f''(x) = 0, this point is not a point of inflexion. The reason is that the sign of f' (x) changes from negative to positive.At x2, we have f' (x) [\ne] 0 and f''(x) = 0. But, x2 is not a stationary point, rather it is a point of inflexion. This because the concavity changes from concave upwards to concave downwards and the sign of f' (x) does not change; it stays positive.
At x3 we have f' (x) = 0 and f''(x) = 0. Here, x3 is both a stationary point and a point of inflexion. This is because the concavity changes from concave upwards to concave downwards and the sign of f' (x) does not change; it stays positive.
See also
- First derivative test
- Second derivative test
- Higher order derivative test
- Fermat's theorem (stationary points)
External link
- [Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio] at cut-the-knot
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