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 sqrt (u^{ 2}) =  u  to rewrite function f as follows
f(x) = sqrt (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
