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UPSC NDA I 2016 Mathematics Solved Paper

Question Numbers: 61-64

Let

f: \mathbb{R} \to \mathbb{R}

be a function such that

f(x) = x^{3} + x^{2} f'(1) + x f''(2) + f'''(3)

for

x \in \mathbb{R}

Question : 64 of 120

Marks: +1, -0

Consider the following:

1. f(2) = f(1) - f(0)

f''(2) - 2f'(1) = 12

Which of the above is/are correct?

Solution:

Concept:

Evaluate polynomial function values and derivatives to verify given conditions.

Explanation:

Given

f(x) = x^3 - 5x^2 + 2x + 6

Compute

f(0) = 6

f(1) = 1 - 5 + 2 + 6 = 4

f(2) = 8 - 20 + 4 + 6 = -2

Then

f(1) - f(0) = 4 - 6 = -2 = f(2)

, so statement 1 is correct.

First derivative:

f'(x) = 3x^2 - 10x + 2

;

f'(1) = 3 - 10 + 2 = -5

Second derivative:

f''(x) = 6x - 10

;

f''(2) = 12 - 10 = 2

Then

f''(2) - 2f'(1) = 2 - 2(-5) = 2 + 10 = 12

, so statement 2 is also correct.

Answer:

Both statements 1 and 2 are correct.

Option C: Both 1 and 2.

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