The intervals in which the function \(f(x) = x^{4} - 18x^{2} + 5\) is strictly increasing are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \((-3, 0)\) and \((3, \infty)\)
Differentiating and factorising:
\( f'(x) = 4x^{3} - 36x \)
\( = 4x(x^{2} - 9) \)
\( = 4x(x + 3)(x - 3) \)
The sign of f′(x) changes at x = −3, 0 and 3:
\( x \lt -3 \): three negative factors, \( f'(x) \lt 0 \)
\( -3 \lt x \lt 0 \): two negative factors, \( f'(x) \gt 0 \)
\( 0 \lt x \lt 3 \): one negative factor, \( f'(x) \lt 0 \)
\( x \gt 3 \): no negative factor, \( f'(x) \gt 0 \)
So f is strictly increasing in (−3, 0) and in \((3, \infty)\).
Why the other options are incorrect
Option (2) \((-\infty, -3)\) and \((0, 3)\) — Incorrect. These are the intervals where f′(x) < 0, so f is decreasing there.
Option (3) \((-\infty, -3)\) and \((3, \infty)\) — Incorrect. f is decreasing on \((-\infty, -3)\), for example f′(−4) = 4(−4)(−1)(−7) = −112 < 0.
Option (4) \((-3, 3)\) only — Incorrect. f decreases on (0, 3), for example f′(1) = 4 − 36 = −32 < 0.