Question

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

• (A) - (IV), (B) - (II), (C) - (I), (D) - (III)
• (A) - (II), (B) - (III), (C) - (I), (D) - (IV)
• (A) - (IV), (B) - (III), (C) - (II), (D) - (I)
• (A) - (IV), (B) - (III), (C) - (I), (D) - (II)

Hint:

When finding the absolute maximum and minimum values of a function on a given interval, be sure to evaluate the function at the endpoints and at any critical points within the interval. Critical points are found by setting the first derivative equal to zero and solving for \( x \). Once you have these values, compare them to determine the absolute maximum and minimum.

Answer 1

The Correct Option is D

Calculate the derivative of \( f(x) = 2x^3 - 9x^2 + 12x - 5 \) to obtain \( f'(x) = 6x^2 - 18x + 12 \).
Set \( f'(x) = 0 \) and solve for \( x \) to identify critical points within the interval \([0, 3]\).
To find the absolute maximum and minimum values, evaluate \( f(x) \) at the interval endpoints \( x = 0 \) and \( x = 3 \), as well as at the identified critical points.