What is a critical number of a function?

A number a in the domain of a function f is called a critical number if either f'(a) = 0 or f'(a) is undefined.

How do you find critical numbers for polynomial functions?

Find the first derivative of the polynomial function, set it equal to zero, and solve for x. All real solutions that lie within the function's domain are critical numbers.

Are points where the derivative is undefined always critical numbers?

They are critical numbers only if the point x = a lies within the actual domain of the original function f.

How to Find Critical Numbers of Functions

Learn how to identify critical numbers of various function types—including polynomials, absolute values, rational expressions, and radicals—using derivatives.

Definition of Critical Numbers

A number \( a \) in the domain of a function \( f \) is called a critical number of \( f \) if either:

\[ f'(a) = 0 \quad \text{or} \quad f' \text{ is undefined at } x = a. \]

Step-by-Step Examples

Example 1: Polynomial function \( f(x) = x^{3} - 3x + 5 \)

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: Absolute value function \( f(x) = |x - 2| \)

The domain is all real numbers. Since \( |u| = \sqrt{u^{2}} \) 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: Graphical derivative analysis

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

Graph of derivative showing zeros and undefined points

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

Example 4: Rational function \( f(x) = \dfrac{x^{2} + 7}{x + 3} \)

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 the numerator equal to zero:

\[ x^{2} + 6x - 7 = 0 \implies (x + 7)(x - 1) = 0 \implies x = -7 \text{ or } x = 1. \]

Although \( f' \) is undefined at \( x = -3 \), \( -3 \) is not in the domain, so it is not a critical number. Therefore, the critical numbers are \( x = -7 \) and \( x = 1 \).

Example 5: Radical function \( f(x) = (x - 2)^{\dfrac{2}{3}} + 3 \)

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:

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

Answers to Exercises

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