The maximum value of function f(x) = 3x³ - 8x² + 27x on the set S = {x ∈ R: x² + 30 ≤ 11x} is:

S = {x ∈ R, x² + 30 ≤ 11x} = {x ∈ R, x² - 11x + 30 ≤ 0} = {x ∈ R, (x - 5)(x - 6) ≤ 0} = {x ∈ R, 5 ≤ x ≤ 6}
Now f(x) = 3x³ - 8x² + 27x ⇒ f'(x) = 9x² - 16x + 27
The discriminant is 16² - 4(9)(27) = 256 - 972 = -716 < 0. Since the leading coefficient is positive, f'(x) > 0 for all x.
Thus f(x) is increasing in [5, 6].
Hence maximum value = f(6) = 3(6)³ - 8(6)² + 27(6) = 3(216) - 8(36) + 162 = 648 - 288 + 162 = 522