For the function \(f(x) = x^{3} - 6x^{2} + 9x + 2\), \(x \in R\):
(A) f is strictly increasing in \((-\infty, 1)\).
(B) f is strictly decreasing in (1, 3).
(C) f is strictly increasing in (2, 4).
(D) f is strictly decreasing in \((3, \infty)\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (B) only
Differentiating and factorising:
\( f'(x) = 3x^{2} - 12x + 9 \)
\( = 3(x - 1)(x - 3) \)
The sign of f′(x) changes at x = 1 and x = 3:
\( x \lt 1 \): \( f'(x) \gt 0 \)
\( 1 \lt x \lt 3 \): \( f'(x) \lt 0 \)
\( x \gt 3 \): \( f'(x) \gt 0 \)
So f is strictly increasing in \((-\infty, 1)\) and \((3, \infty)\), and strictly decreasing in (1, 3).
Checking the statements
(A) f is strictly increasing in \((-\infty, 1)\). — Correct. Both factors are negative, so f′(x) > 0.
(B) f is strictly decreasing in (1, 3). — Correct. The factors have opposite signs, so f′(x) < 0.
(C) f is strictly increasing in (2, 4). — Incorrect. f decreases on (2, 3): f′(2.5) = 3(1.5)(−0.5) < 0.
(D) f is strictly decreasing in \((3, \infty)\). — Incorrect. Both factors are positive there, so f is increasing.