How to Find Critical Numbers of Functions

Tutorial explaining how to identify critical numbers of various types of functions.

Definition of Critical Numbers

A number \( a \) in the domain of a function \( f \) is called a critical number of \( f \) if either \[ f'(a) = 0 \quad \text{or} \quad f' \text{ is undefined at } x = a. \]

Example 1

Find the critical number(s) of the polynomial function \[ f(x) = x^{3} - 3x + 5. \]

Solution

The domain of \( f \) is all real numbers. Its derivative is \[ f'(x) = 3x^{2} - 3. \] Since \( f' \) is defined everywhere, solve \( f'(x) = 0 \): \[ 3x^{2} - 3 = 0 \implies x^{2} = 1 \implies x = \pm 1. \] Both \( x = 1 \) and \( x = -1 \) lie in the domain, so they are critical numbers.

Example 2

Find the critical number(s) of the absolute value function \[ f(x) = |x - 2|. \]

Solution

The domain is all real numbers. Since \[ |u| = \sqrt{u^{2}} \quad \text{where } u = x - 2, \] rewrite \( f \) as \[ f(x) = \sqrt{(x - 2)^{2}}. \] Using the chain rule, the derivative is \[ f'(x) = \dfrac{(x - 2)}{|x - 2|}. \] Note that \( f' \) is undefined at \( x = 2 \). Since \( 2 \) is in the domain, it is a critical number.

Example 3

Find the critical numbers of a function \( f \) whose derivative \( f' \) is shown graphically below.

Graph of derivative showing zeros and undefined points

Solution

The critical numbers are \( x = 1, -2, -3 \) where \( f'(x) = 0 \), and \( x = 0 \) where \( f' \) is undefined.

Example 4

Find the critical numbers of the rational function \[ f(x) = \dfrac{x^{2} + 7}{x + 3}. \]

Solution

The domain is all real numbers except \( x = -3 \). Using the quotient rule, the derivative is \[ f'(x) = \dfrac{2x(x + 3) - (x^{2} + 7)(1)}{(x + 3)^2} = \dfrac{x^{2} + 6x - 7}{(x + 3)^2}. \] Setting numerator equal to zero, \[ x^{2} + 6x - 7 = 0 \implies (x + 7)(x - 1) = 0, \] so \( x = -7 \) or \( x = 1 \). Since \( f' \) is undefined at \( x = -3 \), but \( -3 \notin \) domain, it is not a critical number. Therefore, the critical numbers are \( x = -7 \) and \( x = 1 \).

Example 5

Find the critical number(s) of the function \[ f(x) = (x - 2)^{\dfrac{2}{3}} + 3. \]

Solution

The domain is all real numbers. Its derivative is \[ f'(x) = \dfrac{2}{3}(x - 2)^{-\dfrac{1}{3}} = \dfrac{2}{3(x - 2)^{\dfrac{1}{3}}}. \] The derivative is undefined at \( x = 2 \), which lies in the domain. Thus, \( x = 2 \) is a critical number.

Exercises on Critical Numbers

Find the critical numbers of the following functions:

  1. \( f(x) = 2x^{3} + 6x - 13 \)
  2. \( f(x) = |x + 4| + 3 \)
  3. \( f(x) = (x - 3)^{3} - 5 \)
  4. \( f(x) = x^{\dfrac{1}{3}} + 2 \)
  5. \( f(x) = \dfrac{x}{x + 4} \)

Answers to Exercises

  1. \( 1, -1 \)
  2. \( -4 \)
  3. \( 3 \)
  4. \( 0 \)
  5. No critical numbers

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