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:

For any function, it's increasing where its derivative is positive ($f'(x) > 0$).

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(A) $f(x) = -x^2 - 2x + 1$

Find the derivative:

$f'(x) = -2x - 2$

$f'(x) = -2(x + 1)$

For the function to be increasing:

$-2(x + 1) > 0$

$x + 1 < 0$

$x < -1$

Increasing on $(-\infty, -1)$ → Matches (I)

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(B) $f(x) = x^2 + 1$

Find the derivative:

$f'(x) = 2x$

For the function to be increasing:

$2x > 0$

$x > 0$

Increasing on $(0, \infty)$ → Matches (IV)

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(C) $f(x) = x^2 - 2x + 3$

Find the derivative:

$f'(x) = 2x - 2$

$f'(x) = 2(x - 1)$

For the function to be increasing:

$2(x - 1) > 0$

$x - 1 > 0$

$x > 1$

Increasing on $(1, \infty)$ → Matches (II)

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(D) $f(x) = -x^2$

Find the derivative:

$f'(x) = -2x$

For the function to be increasing:

$-2x > 0$

$x < 0$

Increasing on $(-\infty, 0)$ → Matches (III)

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Final matching:

(A) → (I)

(B) → (IV)

(C) → (II)

(D) → (III)