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.
Solution to Example 1.
The domain of f is the set of all real numbers. The first
derivative f ' is given by
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
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
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 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
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
Simplify to obtain Solving f '(x) = 0 result in solving 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
Solution to Example 5.
Note that the domain of f is the set of all real numbers. The derivative of f is
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.
Answers to Above Exercises.
More on applications of differentiation
