ekstremumide

Extremumide is a term used in mathematics to refer to points where a function attains its maximum or minimum value, either locally or globally. These points are of significant importance in calculus and optimization problems.

A local maximum occurs at a point if the function's value at that point is greater than or equal to the function's values at all nearby points. Similarly, a local minimum occurs when the function's value is less than or equal to the values at all nearby points. These local extrema can be found by examining the first derivative of the function. Critical points, where the first derivative is zero or undefined, are candidates for local extrema.

A global maximum (or absolute maximum) is the highest value the function attains over its entire domain. A global minimum (or absolute minimum) is the lowest value the function attains over its entire domain. Global extrema can occur at local extrema or at the boundaries of the function's domain.

The process of finding extrema often involves using the first derivative test and the second derivative test. The first derivative test helps identify whether a critical point is a local maximum, local minimum, or neither, by analyzing the sign changes of the derivative around the critical point. The second derivative test uses the sign of the second derivative at a critical point to determine the nature of the extremum. If the second derivative is positive, it indicates a local minimum; if it is negative, it indicates a local maximum. If the second derivative is zero, the test is inconclusive.