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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  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Question 18 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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.