Match List-I with List-II
| List-I (Function and interval) | List-II (Behaviour) | ||
| (A) | \(e^{-x}\) on R | (I) | Strictly increasing |
| (B) | \(x^{2}\) on (−1, 1) | (II) | Constant |
| (C) | \(\sin^{2}x + \cos^{2}x\) on R | (III) | Strictly decreasing |
| (D) | \(x^{3} + x\) on R | (IV) | Neither increasing nor decreasing |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
The sign of the derivative on the interval decides the behaviour.
(A) \(e^{-x}\) on R
This matches (III) Strictly decreasing.
\( f'(x) = -e^{-x} \lt 0 \) for every x
(B) \(x^{2}\) on (−1, 1)
This matches (IV) Neither increasing nor decreasing.
\( f'(x) = 2x \): negative on (−1, 0), positive on (0, 1)
(C) \(\sin^{2}x + \cos^{2}x\) on R
This matches (II) Constant.
\( \sin^{2}x + \cos^{2}x = 1 \), so \( f'(x) = 0 \) for every x
(D) \(x^{3} + x\) on R
This matches (I) Strictly increasing.
\( f'(x) = 3x^{2} + 1 \gt 0 \) for every x