Match List-I with List-II
| List-I Function f(x) | List-II Interval for increasing/decreasing of function f(x) | ||
|---|---|---|---|
| (A) | \(f(x) = x|x|\) | (I) | Decreases on (0, ∞) |
| (B) | \(f(x)=x^2+2x-5\) | (II) | Increases on (3, ∞) |
| (C) | \(f(x)=x^2-6x+9\) | (III) | Decreases on (−∞, −1) |
| (D) | \(f(x)=-x^2\) | (IV) | Increases on (−∞, ∞) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
A function increases where \(f'(x) \gt 0\) and decreases where \(f'(x) \lt 0\).
(A) \(f(x) = x|x|\)
This matches (IV) Increases on (−∞, ∞).
\( \displaystyle f(x) = \begin{cases}x^2, & x \ge 0\\-x^2, & x \lt 0\end{cases} \)
\( \displaystyle f'(x) = \begin{cases}2x, & x \gt 0\\-2x, & x \lt 0\end{cases} \)
So \(f'(x) = 2|x| \gt 0\) for every \(x \ne 0\), and f increases on the whole real line.
(B) \(f(x)=x^2+2x-5\)
This matches (III) Decreases on (−∞, −1).
\(f'(x) = 2x + 2 \lt 0\) for \(x \lt -1\)
(C) \(f(x)=x^2-6x+9\)
This matches (II) Increases on (3, ∞).
\(f'(x) = 2x - 6 \gt 0\) for \(x \gt 3\)
(D) \(f(x)=-x^2\)
This matches (I) Decreases on (0, ∞).
\(f'(x) = -2x \lt 0\) for \(x \gt 0\)
So the matching is (A)-(IV), (B)-(III), (C)-(II), (D)-(I).
Note: Some List-I items fit more than one List-II entry. \(x|x|\) increases on the whole real line, so (A) also fits (II); \(x^2+2x-5\) increases for \(x \gt -1\), so (B) also fits (II); \(x^2-6x+9\) decreases for \(x \lt 3\), so (C) also fits (III). The question can still be solved because (IV) fits only (A), and Option (2) is the only option containing (A)-(IV).