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.

The first derivative test is a method used in calculus to analyze the critical points of a function and determine whether they correspond to relative maxima or minima. It is a part of the larger topic of calculus known as optimization, which deals with finding the maximum or minimum values of a function.

Note: The maximum and minimum values are also called the extremum values and the point which is either a point of local maxima or local minima is called an extreme point.

Theorem: If f'(x) exists in the interval [a, b] and the function f(x) has a maximum or minimum value at x = x0 ∈ (a, b) then f'(x0) = 0. The point x₀ in the domain of a function at which f'(x₀) = 0 is called a critical point.

Read More: Differential Calculus

Conditions for First Derivative Test

Let f(x) be a differentiable function defined on an interval I and let x₀ ∈ I and f'(x₀) = 0. Then

• x₀ is a point of the local maximum of f(x) if
  • f'(x₀) changes sign from positive to negative as x increases through x₀ i.e., f'(x) > 0.
• x₀ is a point of local minimum of f(x) if
  • f'(x) changes sign from negative to positive as x increases through x i.e., f'(x) < 0 at every point sufficiently close to x₀ and the left of x and F (x) > 0 at every point sufficiently close to and to the right of x₀.
•  f'(x) does not change sign as x increases through x, then x₀ is neither a point of local maximum nor a point of local minimum. Such a point is called a point of inflection.

Learn more about Inflection Point.

First Derivative Test for Maxima and Minima

Using the first derivative test we can find the local maxima and minima of any given function. Let’s understand maxima and minima first.

Maxima

Let be a real-valued function defined on an interval I. Then f(x) is said to have a maximum value in l at x₀ if x₀ ∈ l and f(x₀) ≥ f (x) for all x in I. The value given for f(x₀) is called the maximum value of f(x).

Minima

Let be a real-valued function defined on an interval I. Then f(x) is said to have a minimum value in I at x₀ if x₀ ∈ I and f(x₀)<=F(x) for all x in I. The value given for f(x₀) is called the minimum value of f(x).

Learn more about Maxima and Minima.

Inflection Point

For any function f(x), the point at which the second derivative is either 0 or doesn’t exist and the sign of the second derivative changes from either positive to negative or negative to positive is called the inflection point.

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.