Match List-I with List-II
| List-I | List-II | ||
| Function | Increasing on the interval | ||
| (A) | \(f(x) = -x^2 - 2x + 1\) | (I) | \((-\infty, -1)\) |
| (B) | \(f(x) = x^2 + 1\) | (II) | \((1, \infty)\) |
| (C) | \(f(x) = x^2 - 2x + 3\) | (III) | \((-\infty, 0)\) |
| (D) | \(f(x) = -x^2\) | (IV) | \((0, \infty)\) |
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(I), (B)-(IV), (C)-(II), (D)-(III)
A function is increasing where \(f'(x) \gt 0\).
(A) \(f(x) = -x^2 - 2x + 1\)
This matches (I) \((-\infty, -1)\).
\(f'(x) = -2x - 2 \gt 0\)
\(-2x \gt 2,\quad x \lt -1\)
(B) \(f(x) = x^2 + 1\)
This matches (IV) \((0, \infty)\).
\(f'(x) = 2x \gt 0,\quad x \gt 0\)
(C) \(f(x) = x^2 - 2x + 3\)
This matches (II) \((1, \infty)\).
\(f'(x) = 2x - 2 \gt 0,\quad x \gt 1\)
(D) \(f(x) = -x^2\)
This matches (III) \((-\infty, 0)\).
\(f'(x) = -2x \gt 0,\quad x \lt 0\)
So the matching is (A)-(I), (B)-(IV), (C)-(II), (D)-(III).