Find all values of c in the open interval (a, b) such that f ′(c) = 0. (Enter your answers as a comma-separated list. If Rolle's theorem cannot be applied, enter NA.)

Question

If Rolle's theorem can be applied, find all values of c in the open interval (a, b)

 such that f ′(c) = 0.

 (Enter your answers as a comma-separated list. If Rolle's theorem cannot be applied, enter NA.)

Answer 1

f(x) = (x-6)(x-7)(x-9)

f is differentiable on (6,9) and is continuous on [6,9] and f(6) = f(9) = 0. So, Rolle's Theorem is applicable.

f'(x) = (x-7)(x-9) + (x-6)(x-9) + (x-6)(x-7) = x² - 16x + 63 + x² - 15x + 54 + x² - 13x + 42

f'(c) = 3c² - 44c + 159 = 0

c = [44 ± √28] / 6 = (22 + √7) / 3 or (22 - √7) / 3 ≈ 8.22, 6.45