What are local extrema?
If a function f(x) has a local maximum, then that means in some local neighborhood of x the function is maximal at x, but not necessarily if you look beyond that local neighborhood.
An extremum (singular of extrema) could also be a local minimum
Local extrema, also known as local maxima and minima, are points on a function where the function reaches a highest or lowest value within a specific interval. These points are significant because they represent the "peaks" and "valleys" on a graph of the function in a certain range. Determining the local extrema is important in many fields, such as optimization problems in mathematics, economics, and engineering.
To find local extrema, follow these steps:
1. Identify the critical points of the function by finding where its derivative is zero or undefined. These points include stationary points (where the derivative is zero) and points where the derivative is undefined (e.g., cusps or vertical tangents).
a. To find the derivative, differentiate the function with respect to the variable in consideration.
2. Determine the intervals between the critical points. These intervals are the ranges over which you will evaluate the function's value.
3. Evaluate the function at the critical points and the endpoints of each interval.
4. Compare the function values at these points to identify the highest and lowest points in each interval. The highest point is the local maximum, while the lowest point is the local minimum. If a point occurs at the end of an interval and has the highest or lowest value among all the points in that interval, it is termed an absolute maximum or minimum.
a. To compare the values, you can use either a table of values or a graph.
Once you have identified the local extrema, you can further analyze the behavior of the function and its properties within the specified interval. Keep in mind that the presence of local extrema does not guarantee the existence of a global (overall) maximum or minimum for the entire function.