If f(x) = x^3 - x^2 f'(1) + xf''(2) - f'''(3), x ∈ R, then — JEE Main 2023 Mathematics PYQ

JEE Main 2023 · 24 Jan · Shift 2

f(x) = x^3 - x^2 f'(1) + xf''(2) - f'''(3)

x \in \mathbb{R}

, then

A.
$f(1) + f(2) + f(3) = f(0)$
B.
$f(3) - f(2) = f(1)$
C.
$3f(1) + f(2) = f(3)$
D.
$2f(0) - f(1) + f(3) = f(2)$

Show answer and solution

Correct option: D

Solution

Let a = f'(1), b = f''(2), c = f'''(3). Then f(x) = x³ - ax² + bx - c.

Differentiating: f'(x) = 3x² - 2ax + b, f''(x) = 6x - 2a, f'''(x) = 6.

From f'''(3) = 6 → c = 6.
From f''(2) = 12 - 2a = b → b = 12 - 2a.
From f'(1) = 3 - 2a + b = a → 3 - 2a + (12 - 2a) = a → 15 = 5a → a = 3, b = 6, c = 6.

So f(x) = x³ - 3x² + 6x - 6.

Computing: f(0) = -6, f(1) = 1-3+6-6 = -2, f(2) = 8-12+12-6 = 2, f(3) = 27-27+18-6 = 12.

Check option 4: 2f(0) - f(1) + f(3) = 2(-6) - (-2) + 12 = -12 + 2 + 12 = 2 = f(2). ✓

[JEE Main 2023 · 24 Jan · Shift 2]

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