What is a stationary point?

A stationary point is defined as a point on a curve where the derivative, or gradient, of a function is zero. This means that at this point, the slope of the tangent line to the curve is horizontal, indicating that the function is neither increasing nor decreasing at that specific point. Stationary points can correspond to local maxima, local minima, or points of inflection, which makes them critical in understanding the behavior of functions in calculus.

In contrast, points where the function is increasing or decreasing are characterized by non-zero gradients, meaning the slope is positive for increasing functions and negative for decreasing functions. A point of discontinuity refers to a location where a function is not continuous; this does not relate to the concept of a stationary point, as stationary points necessitate continuity around them. Therefore, identifying that a stationary point occurs where the gradient is zero accurately captures the essential definition and significance in calculus.