# Find Critical Numbers of Functions

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

f '(x) = [ 2x (x + 3) - (x^{ 2} + 7 )(1) ] / (x + 3)^{ 2}

Simplify to obtain

f '(x) = [ x^{ 2} + 6 x - 7 ] / (x + 3)^{ 2}

Solving f '(x) = 0 result in solving

x^{ 2} + 6 x - 7 = 0

(x + 7)(x - 1) = 0

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)

## Answers to Above Exercises

a) 1 , -1

b) -4

c) 3

d) 0

e) no critical numbers

### More on applications of differentiation

applications of differentiation