Stationary point

Classification

Isolated stationary points of a ${\displaystyle C^1}$ real valued function ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ 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 extrema—are known as saddle points.

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

Curve sketching

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 derivative f″(x):

• 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 ${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$ are those points x₀ where the derivative in every direction equals zero, or equivalently, the gradient is zero.

Examples

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.

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.

For the function f(x) = 0, one has f′(0) = 0 and f″(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 of f′(x) and f″(x) do not change.

The function f(x) = x⁵ sin(1/x) for x ≠ 0, and f(0) = 0, gives an example where f′(x) and f″(x) are both continuous, f′(0) = 0 and f″(0) = 0, and yet f(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.