Steps for First Derivative Test

We can use the following steps to perform the first derivative test.

Step 1: Find the first derivative of the given function.

Step 2: Equate the first derivative to 0, and solve for the dependent variable to find the critical points.

Step 3: With the help of critical points and the domain of the function, find the intervals for the positive and negative slopes of the graph of the function.

Step 4: Check the signs of the derivative around the critical point.

Step 5: Finalize the extremum based on the following cases:

  • If the sign of the first derivative changes from positive to negative as you move from left to right around the critical point, the function has a local maximum at that critical point.
  • If the sign of the first derivative changes from negative to positive as you move from left to right around the critical point, the function has a local minimum at that critical point.
  • If the first derivative does not change sign (either positive or negative) at a critical point, the first derivative test fails and needs to use further tests such as the second derivative test.

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First Derivative: Definition, Formulas, and Examples

First Derivative Test is the test in calculus to find whether a function has a maximum or minimum value in the given interval. As the name suggests, the first derivative is used in this test to find the critical point and then further conditions are used to check each critical point for extrema. Many times in our daily life, we come across many situations where we wish to find the maximum and minimum of something, in those cases, we can use the First Derivative Test if we can mathematically define that situation.

In this article, we will learn this important test i.e., “First Derivative Test”, other than this we will also learn how to use this test to find the extrema of the function as well as solved examples of the same. So, let’s start learning about one of the fundamental tests in calculus i.e., the First Derivative Test.

Table of Content

  • What is First Derivative Test?
  • Steps for First Derivative Test
  • How to Perform First Derivative Test?
  • Example of First Derivative Test

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What is First Derivative Test?

First Derivative Test is a method used to analyze the critical points of a function and determine whether at that point function attains its maximum or minimum value. It is one of the basic tests in calculus which uses the change in slope to determine the extremum value of the function. As we know, either side of each extremum point slope can be either positive or negative....

Steps for First Derivative Test

We can use the following steps to perform the first derivative test....

How to Perform First Derivative Test?

To Perform the First Derivative Test, we can use the following steps:...

Example of First Derivative Test

Example 1: Find the local maximum and local minimum value of the function f(x) = sin x – cos x; 0 < x < 2π, using the first derivative test....

FAQs on First Derivative Test

What is First Derivative Test?...