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.