Stationary point

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function 'stops' increasing or decreasing (hence the name). In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative 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 point is a point on the surface of the graph 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. 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 x ↦ x 3 {displaystyle xmapsto x^{3}} has a stationary point at x=0, which is also an inflection point, but is not a turning point. Isolated stationary points of a C 1 {displaystyle C^{1}} real valued function f : R → R {displaystyle fcolon mathbb {R} o mathbb {R} } are classified into four kinds, by the first derivative test: 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 extremum—are known as saddle points. By Fermat's theorem, global extrema must occur (for a C 1 {displaystyle C^{1}} function) on the boundary or at stationary points. 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):