The interval in which the function \(f(x)=10-6x-2x^2\) is decreasing is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\left(\displaystyle\frac{-3}{2}, \infty\right)\)
A function is decreasing where f′(x) < 0.
\(f'(x) = -6 - 4x\)
\(-6 - 4x \lt 0\)
\(-4x \lt 6\)
Dividing by −4 reverses the inequality:
\( \displaystyle x \gt -\frac{3}{2} \)
So f is decreasing on \(\left(-\frac{3}{2}, \infty\right)\).
Why the other options are incorrect
Option (1) \(\left(\displaystyle\frac{-3}{2}, \frac{3}{2}\right)\) — Incorrect. f is decreasing here, but also beyond \(\frac{3}{2}\); this is only part of the interval.
Option (2) \(\left(-\infty, \displaystyle\frac{-3}{2}\right)\) — Incorrect. Here f′(x) > 0 (for example f′(−2) = 2), so f is increasing; this is the answer if the inequality sign is not reversed.
Option (4) \(\left(-\infty, \displaystyle\frac{3}{2}\right)\) — Incorrect. It contains points where f is increasing, such as x = −2.
NCERT (2026-27 edition):
Increasing and Decreasing Functions
