Let the f : R R be defined by f (x) = 2x + cos x, then f :

(A) has a minimum at x = π         (B) has a maximum, at x = 0

(C) is a decreasing function         (D) is an increasing function

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Question 5 Let the f : R β†’ R be defined by f (x) = 2x + cos x, then f : (A) has a minimum at x = πœ‹ (B) has a maximum, at x = 0 (C) is a decreasing function (D) is an increasing function Given f (𝒙) = 2𝒙 + cos 𝒙 Since we need to check If it is increasing or decreasing Or, at what value of π‘₯ the function has maximum or minimum value Thus, we check sign of 𝒇′(𝒙) ∴ Differentiating 𝑓(π‘₯) w.r.t. π‘₯ f (π‘₯) = 2π‘₯ + cos π‘₯ f’ (𝒙) = 2 βˆ’ sin 𝒙 Checking sign of 𝒇′(𝒙) We know that βˆ’1 ≀ sin 𝒙 ≀ 1 Multiplying by βˆ’1 βˆ’1 Γ— (βˆ’1) β‰₯ βˆ’sin π‘₯ β‰₯ 1 Γ— (βˆ’1) 1 β‰₯ βˆ’ sin π‘₯ β‰₯ βˆ’1 βˆ’1 ≀ βˆ’ sin 𝒙 ≀ 1 Adding 2 both sides 2 βˆ’ 1 ≀ 2 βˆ’ sin π‘₯ ≀ 2 + 1 1 ≀ 2 βˆ’ sin 𝒙 ≀ 3 1 ≀ 𝒇′(𝒙)≀ 3 ∴ 𝒇′(𝒙) > 0 Thus, f (π‘₯) is increasing function So, the correct answer is (D)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.