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UPSC NDA II 2015 Mathematics Paper

Question Numbers: 6-7

For the next two (02) items that follow:

Consider the function

f(x) = -2x^{3} - 9x^{2} - 12x + 1

Question : 6 of 120

Marks: +1, -0

The function f(x) is an increasing in the interval

Solution:

Concept:

f'(x) > 0

, the function is increasing. If

f'(x) < 0

, it is decreasing.

Explanation:

Given

f(x) = -2x^3 - 9x^2 - 12x + 1

Compute the derivative:

f'(x) = -6x^2 - 18x - 12

For

f

to be increasing,

f'(x) > 0

, so

-6x^2 - 18x - 12 > 0

Factor out

-6

-6 (x^2 + 3x + 2) > 0

Multiply both sides by

-1

(reversing inequality):

x^2 + 3x + 2 < 0

Factor the quadratic:

(x+2)(x+1) < 0

The product is negative when

x

lies between the roots

-2

and

-1

Thus,

x \text{ in } (-2, -1)

Answer:

Option A:

(-2, -1)

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