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 1Find the critical number(s) of the polynomial function f given by^{ 3} - 3x + 5
## Solution to Example 1The domain of f is the set of all real numbers. The first derivative f ' is given by^{ 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 2Find the critical number(s) of the absolute value function f given by## Solution to Example 2The domain of f is the set of all real numbers. Let us use the fact √ (u^{ 2}) = | u | to rewrite function f as follows^{ 2}) , with u = x - 2
Using the chain rule, f '(x) is given by Since u '(x) = 1, f '(x) simplifies to 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 3Find the critical number(s) of function f whose first derivative is shown graphically below.## Solution to Example 31, -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 4Find the critical number(s) of the rational function f defined by^{ 2} + 7 ) / (x + 3)
## Solution to Example 4Note that the domain of f is the set of all real numbers except -3. The first derivative of f is given by^{ 2} + 7 )(1) ] / (x + 3)^{ 2}
Simplify to obtain ^{ 2} + 6 x - 7 ] / (x + 3)^{ 2}
Solving f '(x) = 0 result in solving ^{ 2} + 6 x - 7 = 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 5Find the critical number(s) of function f defined by^{ 2/3} + 3
## Solution to Example 5Note that the domain of f is the set of all real numbers. The derivative of f is^{ -1/3}
^{ 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 AnswersFind 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 Exercisesa) 1 , -1 b) -4 c) 3 d) 0 e) no critical numbers
## More on applications of differentiationapplications of differentiation |