For the function \( f(x)=x^{3}-3x^{2}+12x-18 \), which of the following statement is/are correct? Where R is set of real numbers.
(A) \( f'(x)=3\left[(x-1)^{2}+3\right] \)
(B) \( f \) is strictly decreasing on R
(C) \( f \) is strictly increasing on R
(D) \( f \) is neither strictly increasing nor strictly decreasing on R
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
Since \( f(x)=x^{3}-3x^{2}+12x-18 \),
\( \displaystyle f'(x)=3x^{2}-6x+12 \)
\( \displaystyle =3\left(x^{2}-2x+4\right) \)
\( \displaystyle =3\left[(x-1)^{2}+3\right] \)
The bracket is at least 3, so \( f'(x)\ge 9\gt 0 \) for every real x.
So \( f'(x)=3\left[(x-1)^{2}+3\right]\gt 0 \) on R, and f is strictly increasing on R.
Checking the statements
(A) \( f'(x)=3\left[(x-1)^{2}+3\right] \) — Correct. \( 3x^{2}-6x+12=3\left[(x-1)^{2}+3\right] \).
(B) f is strictly decreasing on R — Incorrect. \( f'(x)\ge 9\gt 0 \), so f never decreases.
(C) f is strictly increasing on R — Correct. \( f'(x)\gt 0 \) for every x.
(D) f is neither strictly increasing nor strictly decreasing on R — Incorrect. f is strictly increasing on R.