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Maths Differentiation and Applications of Derivatives General Single Correct MCQ

Published on: August 11, 2026

If ƒ ′′ (x) > 0 and ƒ ′ (1) = 0 such that g(x) = ƒ (cot 2 x + 2 cot x + 2), where 0 < x < π then the interval in which g(x) is decreasing is -

(0, π )

Text Solution

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The correct answer is:

Here, g(x) = ƒ(cot 2 x + 2 cot x + 2)

⇒ g ′ (x) = ƒ ′ (cot 2 x + 2 cot x + 2)

{–2 cot x cosec 2 x – 2cosec 2 x}

For g(x) to be decreasing, g ′ (x) < 0

⇒ ƒ ′ {(cot x + 1) 2 + 1} · (–2 cosec 2 x) (cot x + 1) < 0

⇒ ƒ ′ {(cot x + 1) 2 + 1} · (cot x + 1) > 0 … (i)

{as ƒ ′′ (x) > 0 ⇒ ƒ ′ (x) is increasing, then

ƒ ′{ (cot x + 1) 2 + 1} > ƒ ′ (1) = 0

∀ x ∈ ∪

Thus, equation (i) holds, if cot x + 1 > 0

⇒ cot x > –1 ∀ x ∈

Hence, is the correct answer.

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