Misc 6 - Find intervals in which f(x) = 4 sin x - 2x - x cos x - Miscellaneous

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Misc 6 Find the intervals in which the function f given by f(𝑥) = 4 sin﷮𝑥 − 2𝑥 − 𝑥𝑐𝑜𝑠 𝑥﷯﷮2 + cos﷮𝑥﷯﷯ is (i) increasing (ii) decreasing. f 𝑥﷯ = 4 sin﷮𝑥 − 2𝑥 − 𝑥𝑐𝑜𝑠 𝑥﷯﷮2 + cos﷮𝑥﷯﷯ Let consider the interval 0 , 2𝜋﷯ Step 1: Finding f’ 𝑥﷯ f 𝑥﷯ = 4 sin﷮𝑥 − 2𝑥 − 𝑥𝑐𝑜𝑠 𝑥﷯﷮2 + cos﷮𝑥﷯﷯ = 4 sin﷮𝑥 − 𝑥 2 + cos﷮𝑥﷯﷯﷯﷮2 + cos﷮𝑥﷯﷯ = 4 sin﷮𝑥﷯﷮2 + 𝑐𝑜𝑠﷯ – 𝑥 2 + cos﷮𝑥﷯﷯﷮2 + cos﷮𝑥﷯﷯ = 4 sin﷮𝑥﷯﷮2 + cos﷮𝑥﷯﷯−𝑥 Therefore, f 𝑥﷯= 4 sin﷮𝑥﷯﷮2 + cos﷮𝑥﷯﷯−𝑥 Diff w.r.t 𝑥 f’ 𝑥﷯ = 𝑑﷮𝑑𝑥﷯ 4 sin﷮𝑥﷯﷮2 + 𝑐𝑜𝑠 𝑥﷯ − 𝑥﷯ = 𝑑﷮𝑑𝑥﷯ 4 sin﷮𝑥﷯﷮2 + cos﷮𝑥﷯﷯﷯ – 𝑑 𝑥﷯﷮𝑑𝑥﷯ = 𝑑﷮𝑑𝑥﷯ 4 sin﷮𝑥﷯﷮2 + cos﷮𝑥﷯﷯﷯ – 1 = 4 sin﷮𝑥﷯﷯﷮′﷯ 2+ cos﷮𝑥﷯﷯− 2+ cos﷮𝑥﷯﷯﷮′﷯ 4 sin﷮𝑥﷯﷯﷮ 2+ cos﷮𝑥﷯﷯﷮2﷯﷯﷯ −1 = 4 cos﷮𝑥﷯ 2 + cos﷮𝑥﷯﷯ − − sin﷮𝑥﷯﷯ 4 sin﷮𝑥﷯﷯﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯﷯ −1 = 8 cos﷮𝑥﷯ + 4 cos﷮2﷯﷮𝑥﷯ + 4 sin﷮2﷯﷮𝑥﷯﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯﷯ −1 = 8 cos﷮𝑥﷯ + 4 cos﷮2﷯﷮𝑥﷯ + sin﷮2﷯﷮𝑥﷯﷯﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯−1 = 8 cos﷮𝑥﷯ + 4﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯−1 = 8 cos﷮𝑥﷯ + 4 − 2 + cos﷮𝑥﷯﷯﷮2﷯﷮ 2+ cos﷮𝑥﷯﷯﷮2﷯﷯ = 8 cos﷮𝑥 + 4 − 4 + cos﷮2﷯﷮𝑥 + 4 cos﷮𝑥﷯﷯﷯﷯﷮ 2 + cos﷮𝑥﷯﷮2﷯﷯﷯ = 8 𝑐𝑜𝑠 𝑥 − cos﷮2﷯﷮𝑥﷯ − 4 cos﷮𝑥﷯ ﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯ = 4 cos﷮𝑥 − cos﷮2﷯﷮𝑥﷯﷯﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯ f’ 𝑥﷯ = cos﷮𝑥 4 − cos﷮𝑥﷯﷯﷯﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯ Step 2:Putting f’ 𝑥﷯ = 0 cos﷮𝑥 4 − cos﷮𝑥﷯﷯﷯﷮ 2 + cos﷮𝑥﷯﷯﷮2﷯﷯ = 0 ∴ cos 𝑥 4− cos﷮𝑥﷯﷯ = 0 ∴ cos 𝑥 = 0 𝑥 = 2𝑛+1﷯ 𝜋﷮2﷯ , n ∈ Z Putting n = 0 𝑥 = 2 0﷯+1﷯ 𝜋﷮2﷯ = 𝜋﷮2﷯ Putting n = 1 𝑥 = 2 1﷯+1﷯ 𝜋﷮2﷯ = 3𝜋﷮2﷯ Putting n = 2 𝑥 = 2 2﷯+1﷯ 𝜋﷮2﷯ = 5𝜋﷮2﷯ Since 𝑥 ∈ 0, 2𝜋﷯ So value of 𝑥 are 𝜋﷮2﷯ & 3𝜋﷮2﷯ Step 3: Plotting value of 𝑥 Thus, we divide the interval 0 , 2𝜋﷯ into three disjoint intervals 0 , 𝜋﷮2﷯﷯, 𝜋﷮2﷯ , 3𝜋﷮2﷯﷯ & 3𝜋﷮2﷯,2π﷯ Step 4:

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.