Critical Points and Extrema Value Theorem
Let’s say we have a function f(x), critical points are the points where the derivative of the function becomes zero. These points can either be maxima or minima. A critical point is minima or maxima is determined by the second derivative test. Since there can be more than one point where the derivative of the function is zero, more than minima or maxima is possible. The figure below shows a function that has multiple critical points.
Notice that points A, C are minimas, and points B, D are maximas. B and C are called local maxima and local minima respectively. This means that these points are maximum and minimum in their locality but not necessarily on a global level. Points A and D are called global minima and global maxima.
Let’s say we have a function f(x) which is twice differentiable. Its critical points are given by the f'(x) = 0. Second Derivative Test allows us to check whether the calculated critical point is minima or maxima.
- If f”(x) > 0, then the point x is a maxima.
- If f”(x) < 0, then the point x is a minima.
Now, this test tells us which point is a minimum or a maximum, but it still fails to give us information about the global maxima and global minima. Extrema value Theorem comes to our rescue.
Absolute Minima and Maxima
Absolute Maxima and Minima are the maximum and minimum values of the function defined on a fixed interval. A function in general can have high values or low values as we move along the function. The maximum value of the function in any interval is called the maxima and the minimum value of the function is called the minima. These maxima and minima if defined on the whole functions are called the Absolute Maxima and Absolute Minima of the function.
In this article, we will learn about Absolute Maxima and Mimima, How to calculate absolute maxima and minima, their examples, and others in detail.
Table of Content
- What are Absolute Maxima and Minima?
- Critical Points and Extrema Value Theorem
- Extrema Value Theorem
- Absolute Minima and Maxima in Closed Interval
- Absolute Minima and Maxima in Entire Domain
- What are Local Maxima and Minima?