Local extrema refer to points in a function where the value is either a local maximum or a local minimum. A local maximum is a point where the function value is higher than the values of nearby points, while a local minimum is where the function value is lower than those nearby. These points are important in calculus and optimization, as they can indicate where a function reaches its highest or lowest values within a specific interval. To find local extrema, one often uses the first derivative test. By calculating the derivative of a function and identifying where it equals zero or is undefined, one can locate critical points. Analyzing the sign of the derivative around these points helps determine whether they are local maxima, local minima, or neither. This process is essential in various fields, including mathematics, economics, and engineering.