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 1Find the critical number(s) of the polynomial function f given bySolution to Example 1The domain of f is the set of all real numbers. The first derivative f ' is given byf '(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 bySolution 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 followsUsing 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 bySolution to Example 4Note that the domain of f is the set of all real numbers except -3. The first derivative of f is given bySimplify 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 5Find the critical number(s) of function f defined bySolution to Example 5Note that the domain of f is the set of all real numbers. The derivative of f isf ' 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
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