Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
The given function is:
f(x) = 4x3 - 18x2 + 27x - 7
On differentiating both sides with respect to x, we get
f'(x) = 12x2 - 36x + 27
⇒f'(x) = 3(4x2 - 12x + 9)
⇒f'(x) = 3(2x - 3)2
which is always positive for all x ∈ R.
Since, f'(x) ≥ 0 ∀ x ∈ R,
Therefore, f(x) is always increasing on R