Some Derivative Tests
To find the maxima and minima we use some derivative tests. The following are the two derivative tests to find maxima and minima.
- First Order Derivative Test
- Second Order Derivative Test
First Order Derivative Test
The first order derivative test as the name suggests it uses first order derivative to find maxima and minima. The first order derivative gives the slope of the function.
Let f be a continuous function at critical point c on the open interval l such that f'(c) = 0 then, we will check the nature of the curve. Below are some conditions after checking the nature of the curve, and x increases towards c i.e., the critical point.
- If the sign of f'(x) changes from positive to negative, then f(c) is the maximum value and c is the point of local maxima.
- If the sign of f'(x) changes from negative to positive, then f(c) is the minimum value and c is the point of local minima.
- If the sign of f'(x) neither changes from positive to negative nor from negative to positive, then c is called the point of inflection i.e., neither maxima nor minima.
Read More about First Derivative Test.
Second Derivative Test
The second order derivative test as the name suggests it uses second order derivative to find maxima and minima.
Let f be a function that is two times differentiable at critical point c defined on the open interval l. The following are the conditions:
|
Condition |
Result |
|---|---|
| If f'(c) = 0 and f”(c) < 0 | c is the local maxima and f(c) is the maximum value. |
| If f'(c) = 0 and f”(c) > 0 | c is the local minima and f(c) is the minimum value. |
| If f”(c) = 0 | Test fails. |
Properties of Maxima and Minima
Some properties of the maxima and minima are:
- There is at least one maximum and one minimum that should lie between equal values of f(x), if f(x) is continuous function in its domain.
- There is one maxima in between two minima and vice-versa. Maxima and minima occur alternatively.
- There can only be one absolute maxima and one absolute minima over the entire domain.
Maxima and Minima in Calculus
Maxima and Minima in Calculus is an important application of derivatives. The Maxima and Minima of a function are the points that give the maximum and minimum values of the function within the given range. Maxima and minima are called the extremum points of a function.
This article explores the concept of maxima and minima. In addition to details about maxima and minima, we will also cover the types of maxima and minima, properties of Maxima and Minima, provide examples of maxima and minima, and discuss applications of Maxima and Minima.
Table of Content
- Maxima and Minima of a Function
- Types of Maxima and Minima
- Relative Maxima and Minima
- Absolute Maxima and Minima
- How to Find Maxima and Minima?
- Applications of Maxima and Minima
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