Match List-I with List-II
The function \(f(x) = 2x^3 - 15x^2 +36x +5\) for \(x \in [2,5]\) has
| List-I | List-II | ||
|---|---|---|---|
| (A) | absolute maximum value | (I) | 5 |
| (B) | absolute minimum value | (II) | 60 |
| (C) | point of absolute maxima | (III) | 3 |
| (D) | point of absolute minima | (IV) | 32 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
On a closed interval the absolute extrema occur at critical points or at the end points.
\(f'(x) = 6x^2 - 30x + 36 = 6(x - 2)(x - 3)\)
So the critical points are x = 2 (also an end point) and x = 3. The values to compare are
\(f(2) = 16 - 60 + 72 + 5 = 33\)
\(f(3) = 54 - 135 + 108 + 5 = 32\)
\(f(5) = 250 - 375 + 180 + 5 = 60\)
(A) absolute maximum value
This matches (II) 60, the largest of 33, 32 and 60.
(B) absolute minimum value
This matches (IV) 32, the smallest of the three values.
(C) point of absolute maxima
This matches (I) 5, since the maximum value 60 is f(5).
(D) point of absolute minima
This matches (III) 3, since the minimum value 32 is f(3).
So the matching is (A)-(II), (B)-(IV), (C)-(I), (D)-(III).