Defining an Extremum
An extremum (plural: extrema) in mathematics refers to a maximum or minimum value of a function. These points represent the highest (maximum) or lowest (minimum) values a function can attain, either across its entire domain (global extrema) or within a specific interval (local extrema).
Types of Extrema: Local vs. Global
There are two main types: local (or relative) and global (or absolute) extrema. A local maximum is a point where the function's value is greater than or equal to values at nearby points, while a local minimum is where it's less than or equal. A global maximum or minimum is the absolute highest or lowest value over the entire domain of the function.
Identifying Extrema Using Calculus
In calculus, extrema of differentiable functions are typically found by identifying critical points where the first derivative of the function is zero or undefined. These critical points are candidates for local maxima or minima, which can then be further classified using the first or second derivative tests.
Practical Importance and Applications
Understanding extrema is fundamental in solving optimization problems across diverse fields such as engineering, economics, physics, and computer science. For instance, engineers might use this concept to design structures for maximum strength, economists to maximize profit, and scientists to model systems that achieve peak or lowest states.