Let 𝑓(𝑥)=(𝑥−2)2+4𝑥−4. Find its critical number and critical value. If the function has no critical number, enter DNE for both answers.

Question

Let 𝑓(𝑥)=(𝑥−2)2+4𝑥−4. Find its critical number and critical value. If the function has no critical number, enter DNE for both answers.

Answer 1

Critical points occur where f'(x) = 0 or f'(x) = DNE.

We have

f(x) = (x-2)2+4x-4 = 2x-4+4x-4 = 6x - 8

Thus, we have

f'(x) = 6

Notice that f'(x) is never 0. Hence, f(x) does not have any critical points.