Match the LIST-I with LIST-II
| LIST-I | LIST-II |
| Curve \( y=f(x) \), where \( f(x) \) is | Nature / derivative / value of \( y=f(x) \) |
| A. \( x|x| \) | I. \( y' \) is 2 for all \( x\in R^+ \)( set of positive real numbers) |
| B. \( x+|x| \) | II. \( y' \) is 2 for all \( x\in R^- \)( set of negative real numbers) |
| C. \( x-|x| \) | III. \( f(-5)=-1 \) |
| D. \( \displaystyle\frac{x}{|x|} \) | IV. (0, 0) is the point of inflection |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A-IV, B-I, C-II, D-III
Each function is written without the modulus on either side of 0.
(A) \( x|x| \)
This matches IV. f(x) = x² for x ≥ 0 and −x² for x < 0. The curve is concave down on the left of 0 and concave up on the right, so (0, 0) is a point of inflection.
(B) \( x+|x| \)
This matches I. For x > 0, x + |x| = 2x, so y′ = 2.
(C) \( x-|x| \)
This matches II. For x < 0, x − |x| = 2x, so y′ = 2.
(D) \( \displaystyle\frac{x}{|x|} \)
This matches III. For x < 0, x/|x| = −1, so f(−5) = −1.
So the matching is A-IV, B-I, C-II, D-III.