Question

The function \(f(x) = x^3 - 3x^2 + 12x - 18 \) is

(A) strictly decreasing on R

(B) strictly increasing on R

(C) neither strictly increasing nor strictly decreasing on R

(D) strictly decreasing on \((-\infty, 0)\)

Answer 1

Correct option is (B) strictly increasing on R

\(f(x) = x^3 - 3x^2 + 12 x - 18\)

\(f'(x) = 3x^2 - 6x +12 = 3(x^2 - 2x +4)\)

Case I: f is strictly increasing (f'(x) > 0)

\(3(x^2 - 2x + 4) > 0 \rightarrow x^2 - 2x+ 4>0 \rightarrow x^2 - 2x + 1 + 3 > 0\)

\((x-1)^2 + 3> 0\)

\((x-1)^2 > -3 \rightarrow\) Always True because positive real number is always greater than negative real number.