Tutorial on how to find the critical numbers of a function.

Definition

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

Example 1

Find the critical number(s) of the polynomial function f given by

f(x) = x^{ 3} - 3x + 5

Solution to Example 1

The domain of f is the set of all real numbers. The first
derivative f ' is given by

f '(x) = 3 x^{ 2} - 3

f '(x) is defined for all real numbers. Let us now solve f '(x) = 0
3 x^{ 2} - 3 = 0
x = 1 or x = -1
Since x = 1 and x = -1 are in the domain of f they are both critical numbers.

Example 2

Find the critical number(s) of the absolute value function f given by

f(x) = | x - 2 |

Solution to Example 2

The domain of f is the set of all real numbers. Let us use the fact √ (u^{ 2}) = | u | to rewrite function f as follows

f(x) = √ (u^{ 2}) , with u = x - 2

Using the chain rule, f '(x) is given by

f '(x) = (1/2) 2 u u'(x) / | u |

Since u '(x) = 1, f '(x) simplifies to

f '(x) = (x - 2) / | x - 2 |

f ' is undefined at x = 2 and 2 is in the domain of f. x = 2 is a critical number of function f given above.

Example 3

Find the critical number(s) of function f whose first derivative is shown graphically below.

Solution to Example 3

1, -2 ,-3 and 0 are critical numbers since f '(x) is equal to 0 at x = 1, -2, -3 and is undefined at x = 0

Example 4

Find the critical number(s) of the rational function f defined by

f(x) = (x^{ 2} + 7 ) / (x + 3)

Solution to Example 4

Note that the domain of f is the set of all real numbers except -3. The first derivative of f is given by

x = -7 or x = 1
f '(x) is undefined at x = -3 however x = -3 is not included in the domain of f and cannot be a critical number. The only critical numbers of f are x = -7 and x = 1.

Example 5

Find the critical number(s) of function f defined by

f(x) = (x - 2)^{ 2/3} + 3

Solution to Example 5

Note that the domain of f is the set of all real numbers. The derivative of f is

f '(x) = (2/3)(x - 2)^{ -1/3}

= 2 / [ 3(x - 2)^{ 1/3}]

f ' is undefined at x = 2 and since x = 2 is in the domain of f it is a critical number.

Exercises on Critical Numbers With Answers

Find the critical numbers of the functions:
a) f(x) = 2x^{ 3} + 6 x - 13
b) f(x) = | x + 4 | + 3
c) f(x) = (x - 3)^{ 3} - 5
d) f(x) = x^{ 1/3} + 2
e) f(x) = x / (x + 4)