Maxima and Minima Concept: Meaning, Types, and Identification

Maxima and Minima Concept: Maxima and minima are the extreme points of the function, that is, the highest and the lowest points, respectively. For a given set of ranges, the maxima is the maximum value of a function and the minima is the lowest value of a function. The maximum value of a function under the complete range can also be called the absolute maxima and the minimum value of a function under the complete range can also be called the absolute minima.

Functions can have other different types of maxima and minima as well. These are the local maxima and local minima of functions. Please continue reading this article to know more about the concept of absolute maxima and minima, local maxima and minima, methods to identify the maxima and minima of a function, and more.

Meaning of the Maxima and Minima of Functions

Every curve of a function has a peak and a valley that are the maxima and minima, respectively. A function does not have a fixed number of maxima and minima. The maxima and minima in Calculus can be found without even glancing at the graph of that function. The highest point of the curve of a particular range is the maxima and the lowest point of the curve is the minima. Maxima and minima, combined together, form the extrema.

Maxima and minima are of two types. They are absolute maxima and absolute minima, and local maxima and local minima. Absolute maxima is also known as global minima and similarly, absolute minima is also known as global minima.

What are Absolute Maxima and Absolute Minima?

The absolute maxima of a function is the highest point in the entirety of that particular domain of that function. On the other hand, the absolute minima is the lowest point in the entirety of that particular domain of that function. Over the entire domain, there can only exist one absolute maxima and one absolute minima.

One can also refer to the absolute maxima and minima of a function as global maxima and minima, respectively.

What are Local Maxima and Local Minima?

For a particular interval, the maxima of the function is called the local maxima and the minima of the function is called the local minima. The value that a function can hold at any one point over a particular interval that is lesser than the function value at that point is the local maxima. Conversely, the value that a function can hold at any one point over a particular interval that is greater than the function value at that point is the local minima.

Identifying the Maxima and Minima of Functions

The maxima and minima of a function can be found using two derivative tests. The first-order derivative test and the second-order derivative test. Please keep reading to know about both these derivative tests.

First-Order Derivative Test

The slope of the function is provided by the first derivative of a function. The slope of a curve near a maximum point becomes greater toward the maximum point, then reaches 0 at the maximum point, and then becomes smaller when moving away from the maximum point. Likewise, the slope of a curve near the minimum point becomes smaller toward the minimum point, then reaches 0 at the minimum point, and then becomes greater when moving away from the minimum point.

Local maxima: If f’(x) switches from positive to negative when x increases over point c, the maximum value of the function of that particular range can be given by f(c).

Local minima: If f’(x) switches from negative to positive when x increases over point c, the minimum value of the function of that particular range can be given by f(c).

Inflection Point: If f’(x) does not switch signs when x increases over c, and c is neither the maxima or minima of that particular function, then c is the inflection point.

Second-Order Derivative Test

For the second-order derivative test, the first derivative of a particular function is obtained and if that provides the slope value that is equal to 0 at the critical point x = c(f’(c) = 0), only then can the second-order derivative be obtained for that particular function.

If within that particular range, there exists the second derivative of the function, then that given point is the local maxima, if f’’(c) < 0; or is the local minima, if f’’(c) > 0; or if the test fails, that means, f’’(c) = 0.

These are the methods of identifying maxima and minima of functions.

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