For the function \(f(x) = 2x^3 - 9x^2 + 12x - 5,\ x \in [0, 3]\), match List-I with List-II:
| List-I | List-II | ||
| (A) | Absolute maximum value | (I) | 3 |
| (B) | Absolute minimum value | (II) | 0 |
| (C) | Point of maxima | (III) | −5 |
| (D) | Point of minima | (IV) | 4 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (IV), (B) - (III), (C) - (I), (D) - (II)
On a closed interval, the absolute maximum and minimum occur at a critical point or at an end point. Critical points:
\(f'(x) = 6x^2 - 18x + 12 = 6(x^2 - 3x + 2)\)
\(6(x-1)(x-2) = 0\)
\(x = 1, 2\)
Values at the critical points and the end points 0 and 3:
\(f(0) = -5\)
\(f(1) = 2 - 9 + 12 - 5 = 0\)
\(f(2) = 16 - 36 + 24 - 5 = -1\)
\(f(3) = 54 - 81 + 36 - 5 = 4\)
(A) Absolute maximum value
This matches (IV) 4. The largest of the four values is f(3) = 4. The value f(1) = 0 is only a local maximum.
(B) Absolute minimum value
This matches (III) −5. The smallest value is f(0) = −5.
(C) Point of maxima
This matches (I) 3. The absolute maximum is reached at x = 3.
(D) Point of minima
This matches (II) 0. The absolute minimum is reached at x = 0.
So the matching is (A) - (IV), (B) - (III), (C) - (I), (D) - (II).