Final answer:
The function g(x) = 8 - 2tan(x) has no critical numbers where the derivative is zero, but it is undefined at x = π/2 + nπ, where n is any integer, so the proper response is DNE (Does Not Exist).
Step-by-step explanation:
To find the critical numbers of the function g(x) = 8 - 2tan(x), we need to determine where the derivative of the function is zero or undefined. The derivative of tan(x) is sec2(x), so the derivative of g(x) is given by:
g'(x) = -2sec2(x). Since sec2(x) is never zero, there are no critical points where g'(x) = 0. However, sec(x) is undefined when x is an odd multiple of π/2, therefore the critical numbers are those values of x where:
x = π/2 + nπ, where n is any integer and the function is undefined. The critical values of g(x) are thus cannot be listed explicitly as real numbers because the tan function has vertical asymptotes at these odd multiples of π/2. Thus our answer for critical numbers is DNE (Does Not Exist).