Stationary point

In mathematics, particularly in calculus, a stationary point or critical point of a differentiable function of one variable is a point of the domain of the function where the derivative is zero (equivalently, the slope of the graph at that point is zero). It is a point where the function "stops" increasing or decreasing (hence the name).

For a differentiable function of several variables, a stationary (critical) point is an input (one value for each variable) where all the partial derivatives are zero (equivalently, the gradient is zero).

The 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 (more specifically, parallel to the x-axis). For function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the xy plane.

Stationary points, critical points and turning points

The term stationary point of a function may be confused with critical point for a given projection of the graph of the function. "Critical point" is more general: a stationary point of a function corresponds to a critical point of its graph for the projection parallel to the x-axis. On the other hand, the critical points of the graph for the projection parallel to the y axis are the points where the derivative is not defined (more exactly tends to the infinity). It follows that some authors call "critical point" the critical points for any of these projections.