The function f(x) = x2 − 2x is strictly decreasing in the interval
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((-\infty, 1)\)
A function is strictly decreasing where its derivative is negative.
\(f'(x) = 2x - 2 = 2(x - 1)\)
\(f'(x) \lt 0 \text{ when } x \lt 1\)
So f is strictly decreasing in \((-\infty, 1)\), and strictly increasing in \((1, \infty)\), where f’(x) > 0.
Why the other options are incorrect
Option (1) \((1, \infty)\) — Incorrect. Here f’(x) > 0, so f is increasing there.
Option (2) \((-1, \infty)\) — Incorrect. It contains x > 1, where f is increasing; for example f(2) = 0 > f(1) = −1.
Option (4) \((-\infty, 2)\) — Incorrect. It contains 1 < x < 2, where f’(x) > 0.