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.

graph of derivative, example 3

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