First Derivative Test

The first derivative test enables you to verify whether a critical point is a relative minimum, a relative maximum, or neither.

The first derivative test is based on the fact that if a critical point is a relative maximum, the function will be increasing before it and decreasing after it, and if it is a relative minimum, then the function will be decreasing before it and increasing after it.

Further, if a function is increasing on both sides of a critical point, or decreasing on both sides of it, then that critical point is neither a relative maximum nor a minimum.

Method of doing the First derivative test

Suppose there is a function y = f(x) and it has a critical point at x = c. In order to check whether x = c is a relative maximum or minimum by the help of the first derivative test, you need to:

• Find the derivative of f(x)
• Take an x value just lesser than x = c and evaluate the derivative of f(x) at that value.
• Take an x value just larger than x = c and evaluate the derivative of f(x) at that value.

Since the first derivative of a function is its rate of change, the values that you calculated above in step 2 and 3 are going to tell you where the function is increasing/decreasing. Now applying these rules:

• If the first derivative is negative, the function is decreasing
• If the first derivative is positive, the function is increasing
• If the first derivative is zero, the function is neither increasing nor decreasing

Using the above rules and comparing the values obtained in the previous step to them, you can get either of these four conditions, whose results are given along with them:

• x = c is a relative maximum: the function is increasing before x = c and decreasing (or constant) after x = c
• x = c is a relative maximum: the function is decreasing (or constant) before and increasing after x = c
• x = c is neither a maximum nor a minimum: the function is either increasing or decreasing on both sides of x = c, or the function is constant on either side.

Example:

For the function `f(x) = x^3 + 3x^2 - 4x + 1`,

• First derivative, `f'(x) = 3x^2 + 6x - 4`
• Critical points: -2.53, 0.53

Now take a value of x between the critical points, and before and after the first and last critical point, and evaluate the derivative at these values of x:
Values of x:

• before -2.53: -3 
• between -2.53 and 0.53: 0
• after 0.53: 1

Graph of f(x) displays the relative extremes

The first derivative at each of these x values evaluates as follows:

• f'(-3) = 5
• f'(0) = -4
• f'(1) = 5

Remembering the rule that when the first derivative is positive then the function is increasing, and when it is negative the function is decreasing, and keeping in mind the intervals in which each of the above three x values lie, we have:

• Interval: `(-\infty, -2.53)` : Function is increasing
• Interval: (-2.53, 0.53) : Function is decreasing
• Interval: `(0.53, \infty)` : Function is increasing