stasjonærpunkt

A stasjonærpunkt, or stationary point, is a point on the graph of a function where the derivative of the function is zero. This means that at a stationary point, the tangent line to the graph is horizontal. These points are significant because they often correspond to local maxima, local minima, or inflection points of the function.

To find stationary points, one must first compute the derivative of the function. Then, the derivative is set equal to zero, and the resulting equation is solved for the independent variable. The x-values obtained are the locations of the stationary points. To determine the nature of these points, further analysis is usually required, often involving the second derivative test.

The second derivative test checks the concavity of the function at the stationary point. If the second derivative is positive, the point is a local minimum. If it is negative, the point is a local maximum. If the second derivative is zero, the test is inconclusive, and other methods, such as examining the sign of the first derivative on either side of the point, must be used to classify it as a saddle point or an inflection point. Stationary points are a fundamental concept in calculus for understanding the behavior and critical features of functions.