First Derivative Test

What is First Derivative Test?

First Derivative Test is the method in calculus which help us analyze the behaviour of any function near the critical point.

How Does the First Derivative Test Work?

First Derivative Test uses the change in the sign of the derivative near the critical point such that by analysing the change in the sign of the derivative around the critical point, we can conclude that the function has either relative maxima, relative minima, or points of inflection at the critical point.

What are Critical Points?

Critical points are the points on a function where its derivative is either zero or undefined.

What happens if the Derivative is Zero at a Critical Point?

If derivative of any function became zero at the critical point, then the first derivative test become inconclusive in determining the relative maximum, relative minimum, or a point of inflection of the function. In such cases, we need further tests such as Second Derivative Test.

Can the First Derivative Test determine Absolute Extrema?

No, the First Derivative Test can only determine relative extrema. To find absolute extrema, you need to consider the function’s values at the critical points and endpoints within a specific interval.

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.