Match the LIST-I with LIST-II
| LIST-I Function \( f(x) \) | LIST-II \( f(x) \) is increasing in the interval | ||
| A. | \( f(x)=-x^{2}-2x+3 \) | I. | \( (-\infty,2) \) |
| B. | \( f(x)=x^{2}+4x+1 \) | II. | \( (1,\infty) \) |
| C. | \( f(x)=2x^{2}-4x+3 \) | III. | \( (-\infty,-1) \) |
| D. | \( f(x)=-3x^{2}+12x+1 \) | IV. | \( (-2,\infty) \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A-III, B-IV, C-II, D-I
A function is increasing where f′(x) > 0.
(A) \( f(x)=-x^{2}-2x+3 \)
\( f'(x)=-2x-2\gt0 \Rightarrow x\lt-1 \), which matches (III).
(B) \( f(x)=x^{2}+4x+1 \)
\( f'(x)=2x+4\gt0 \Rightarrow x\gt-2 \), which matches (IV).
(C) \( f(x)=2x^{2}-4x+3 \)
\( f'(x)=4x-4\gt0 \Rightarrow x\gt1 \), which matches (II).
(D) \( f(x)=-3x^{2}+12x+1 \)
\( f'(x)=-6x+12\gt0 \Rightarrow x\lt2 \), which matches (I).
So the matching is (A)-(III), (B)-(IV), (C)-(II), (D)-(I).