Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | \( y=x^2 \) | I. | decreasing on \( (0,\infty) \) |
| B. | \( y=-x^2 \) | II. | increasing on \( (0,\infty) \) |
| C. | y = x | III. | decreasing on \( (-\infty,\infty) \) |
| D. | y = −x | IV. | increasing on \( (-\infty,\infty) \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A-II, B-I, C-IV, D-III
A function is increasing where \( \frac{dy}{dx}\gt 0 \) and decreasing where \( \frac{dy}{dx}\lt 0 \), so the sign of the derivative decides each match.
(A) \( y=x^2 \)
This matches II. increasing on (0, ∞). \( \frac{dy}{dx}=2x\gt 0 \) for x > 0.
(B) \( y=-x^2 \)
This matches I. decreasing on (0, ∞). \( \frac{dy}{dx}=-2x\lt 0 \) for x > 0.
(C) y = x
This matches IV. increasing on (−∞, ∞). \( \frac{dy}{dx}=1\gt 0 \) for every x.
(D) y = −x
This matches III. decreasing on (−∞, ∞). \( \frac{dy}{dx}=-1\lt 0 \) for every x.
So the matching is A-II, B-I, C-IV, D-III.