Stationary Points

Stationary Points

A stationary point is a point on a curve where the derivative is zero. The tangent at the stationary point is parallel to the x-axis. There are two types of stationary points, stationary points of inflexion and turning points. For turning points, there are two types, maximum and minimum.

Maximum Turning Point

A maximum turning point is one where, in its neighbourhood, the derivative decreases from positive values, to zero at the turning point, then to negative values. An example is the point (0,0) in the equation y = -x².

In the region x < 0, the derivative is positive but decreasing. When x = 0, the maximum turning point, the derivative is zero. In the region x > 0, the derivative is decreasing and negative.

Minimum Turning Point

Conversely, a minimum turning point is one where, in its neighbourhood, the derivative increases from negative values to zero, then to positive values. Take the example (0,0) on the equation y = x².

In the region x < 0, the derivative is negative but increasing. At x = 0, the minimum turning point, the derivative is zero. In the region x > 0, the derivative is increasing and positive.

Stationary Points of Inflexion

A stationary point of inflexion is one at which the curvature changes. In its neighbourhood, the derivative does not change sign. An example is the point (0,0) on the equation y = x³.

In the region x < 0, the derivative is positive but decreasing. At x = 0, the stationary point of inflexion, the derivative is zero. In the region x > 0, the derivative is still positive but increasing.

Note that there exist non-stationary points of inflexion. There are points at which the curvature of the graph changes, but the derivative is not zero. An example is the point (0,0) on the tangent graph.

The next two sections explain how to determine the nature of stationary points using differentiation. Note that these methods are not limited to the cartesian plane; they are also used in practical problems such as areas and volumes.