Extremal
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). 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 (critical) point is an input (one value for each variable) 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.
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.
A turning point is a point at which the derivative changes sign. 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 has a stationary point at x=0, which is also an inflection point, but is not a turning point.
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