Question

Let f: R → R be a function such that f(x) = x³ + x² f'(1) + xf''(2) + f'''(3), x ∈ R. Then f(2) equals:
1. -4
2. 30
3. -2
4. 8

Answer 1

Correct Answer - Option 3 : -2

From question, the equation given is:

f(x) = x³ + x² f'(1) + xf''(2) + f'''(3)

On differentiating,

⇒ f'(x) = 3x² + 2xf'(1) + f''(2)     ----(1)

Again differentiating,

⇒ f''(x) = 6x + 2f'(1)     ----(2)

Again differentiating,

⇒ f'''(x) = 6     ----(3)

Now,

∴ f'''(3) = 6

On putting x = 1 in equation (1),

⇒ f'(1) = 3 + 2f'(1) + f''(2)     ----(4)

On putting x = 2 in equation (2),

⇒ f''(2) = 12 + 2f'(1)     ----(5)

On substituting equation (5) in equation (4),

⇒ f'(1) = 3 + 2f'(1) + (12 + 2f'(1))

⇒ 3f'(1) = -15

∴ f'(1) = -5

On substituting above value in equation (5),

⇒ f''(2) = 12 + 2(-5)

∴ f''(2) = 2

Now, the given equation becomes,

⇒ f(x) = x³ – 5x² + 2x + 6

⇒ f(2) = 2³ - 5(2)² + 2(2) + 6

⇒ f(2) = 8 – 20 + 4 + 6

∴ f(2) = -2