Match List-I with List-II
| List-I | List-II | ||
| (A) | The interval in which \( f(x)=x^{2}-4x+17 \) is increasing. | (I) | \( (-5,\infty) \) |
| (B) | The interval in which \( f(x)=x^{2}-4x+17 \) is decreasing. | (II) | \( (-\infty,-5) \) |
| (C) | The interval in which \( f(x)=3x^{2}+30x-7 \) is increasing. | (III) | \( (2,\infty) \) |
| (D) | The interval in which \( f(x)=3x^{2}+30x-7 \) is decreasing. | (IV) | \( (-\infty,2) \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
A function is increasing where \( f'(x)\gt0 \) and decreasing where \( f'(x)\lt0 \).
For \( f(x)=x^{2}-4x+17 \):
\( \displaystyle f'(x)=2x-4=2(x-2) \)
For \( f(x)=3x^{2}+30x-7 \):
\( \displaystyle f'(x)=6x+30=6(x+5) \)
(A) The interval in which \( f(x)=x^{2}-4x+17 \) is increasing.
This matches (III) \( (2,\infty) \): \( 2(x-2)\gt0 \) for \( x\gt2 \).
(B) The interval in which \( f(x)=x^{2}-4x+17 \) is decreasing.
This matches (IV) \( (-\infty,2) \): \( 2(x-2)\lt0 \) for \( x\lt2 \).
(C) The interval in which \( f(x)=3x^{2}+30x-7 \) is increasing.
This matches (I) \( (-5,\infty) \): \( 6(x+5)\gt0 \) for \( x\gt-5 \).
(D) The interval in which \( f(x)=3x^{2}+30x-7 \) is decreasing.
This matches (II) \( (-\infty,-5) \): \( 6(x+5)\lt0 \) for \( x\lt-5 \).
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).