The function \(f(x) = x^2 - 4x + 6\) is
(A) Strictly decreasing on \((-\infty, 2) \cup (2, \infty)\)
(B) Strictly increasing on \((2, \infty)\)
(C) Strictly increasing on \((-\infty, \infty)\)
(D) Strictly decreasing on \((-\infty, 2)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
The sign of the derivative decides where f increases or decreases:
\( \displaystyle f'(x) = 2x - 4 = 2(x - 2) \)
\( \displaystyle f'(x) \lt 0 \) for \(x \lt 2\)
\( \displaystyle f'(x) \gt 0 \) for \(x \gt 2\)
So f is strictly decreasing on \((-\infty, 2)\) and strictly increasing on \((2, \infty)\).
Checking the statements
(A) Strictly decreasing on \((-\infty, 2) \cup (2, \infty)\) — Incorrect. f is increasing on \((2, \infty)\).
(B) Strictly increasing on \((2, \infty)\) — Correct. \(f'(x) \gt 0\) there.
(C) Strictly increasing on \((-\infty, \infty)\) — Incorrect. f is decreasing on \((-\infty, 2)\).
(D) Strictly decreasing on \((-\infty, 2)\) — Correct. \(f'(x) \lt 0\) there.
NCERT (2026-27 edition):
Increasing and Decreasing Functions
