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:

The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ on interval $[2, 5]$ needs to be analyzed for absolute maximum and minimum values, and their corresponding x-coordinates.

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Finding the derivative:

$f'(x) = 6x^2 - 30x + 36$

Setting $f'(x) = 0$:

$6x^2 - 30x + 36 = 0$

$x^2 - 5x + 6 = 0$

$(x - 2)(x - 3) = 0$

Critical points: $x = 2$ and $x = 3$

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Evaluating the function at critical points and endpoints:

At $x = 2$:

$f(2) = 2(2)^3 - 15(2)^2 + 36(2) + 5$

$f(2) = 2(8) - 15(4) + 72 + 5$

$f(2) = 16 - 60 + 72 + 5$

$f(2) = 33$

At $x = 3$:

$f(3) = 2(3)^3 - 15(3)^2 + 36(3) + 5$

$f(3) = 2(27) - 15(9) + 108 + 5$

$f(3) = 54 - 135 + 108 + 5$

$f(3) = 32$

At $x = 5$:

$f(5) = 2(5)^3 - 15(5)^2 + 36(5) + 5$

$f(5) = 2(125) - 15(25) + 180 + 5$

$f(5) = 250 - 375 + 180 + 5$

$f(5) = 60$

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Comparing the values:

At $x = 2$: $f(2) = 33$

At $x = 3$: $f(3) = 32$ (minimum)

At $x = 5$: $f(5) = 60$ (maximum)

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The matches are:

(A) Absolute maximum value $= 60$ matches with (II)

(B) Absolute minimum value $= 32$ matches with (IV)

(C) Point of absolute maxima $= 5$ matches with (I)

(D) Point of absolute minima $= 3$ matches with (III)

Therefore, the correct answer is: (A) - (II), (B) - (IV), (C) - (I), (D) - (III)