For the function \(f(x) = x^3 - 3x^2 - 9x + 2\), consider the following statements:
(A) f is strictly increasing in \((-\infty, -1)\).
(B) f is strictly decreasing in \((-1, 3)\).
(C) f is strictly increasing in (0, 4).
(D) \(f''(1) = 0\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
Differentiating:
\(f'(x) = 3x^2 - 6x - 9\)
\(f'(x) = 3(x^2 - 2x - 3)\)
\(f'(x) = 3(x + 1)(x - 3)\)
\(f''(x) = 6x - 6\)
The sign of f′(x): both factors are negative for x < −1, they have opposite signs for −1 < x < 3, and both are positive for x > 3.
So f is strictly increasing in \((-\infty, -1)\) and \((3, \infty)\), strictly decreasing in \((-1, 3)\), and \(f''(1) = 0\).
Checking the statements
(A) f is strictly increasing in \((-\infty, -1)\). — Correct. f′(x) > 0 there, for example f′(−2) = 15.
(B) f is strictly decreasing in \((-1, 3)\). — Correct. f′(x) < 0 there, for example f′(0) = −9.
(C) f is strictly increasing in (0, 4). — Incorrect. f is decreasing in (0, 3), since f′(0) = −9 < 0.
(D) \(f''(1) = 0\). — Correct. \(f''(1) = 6 - 6 = 0\).