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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 : 7 of 120

Marks: +1, -0

The function f(x) is a decreasing in the interval

Solution:

Concept:

A function is decreasing where its derivative is negative:

f'(x) < 0

Explanation:

Given

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

Differentiate:

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

Factor out

-6

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

For decreasing,

f'(x) < 0

, so

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

Divide by

-6

(reverse inequality):

x^2 + 3x + 2 > 0

Factor:

(x+2)(x+1) > 0

This quadratic is positive when both factors are positive or both negative.

Thus,

x < -2

x > -1

So the function is decreasing in

(-\infty, -2) \cup (-1, \infty)

Answer:

Option D:

(-\infty, -2) \cup (-1, \infty)

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