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Ex 6.2, 9 - Prove that y = 4 sin/2 + cos - theta is increasing - To show increasing/decreasing in intervals

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Ex 6.2,9 Prove that 𝑦 = ﷐4﷐sin﷮𝜃﷯﷮(2 +﷐ cos﷮𝜃)﷯﷯ – θ is an increasing function of θ in﷐0,﷐𝜋﷮2﷯﷯ . 𝑦 = ﷐4﷐sin﷮𝜃﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯﷯−𝜃 We need to prove that function is increasing function of θ in ﷐0 , ﷐𝜋﷮2﷯﷯ i.e. we need to prove that ﷐𝑑𝑦﷮𝑑𝜃 ﷯ > 0 for θ ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ We have 𝑦 = ﷐4﷐sin﷮𝜃﷯﷮2 + ﷐cos﷮𝜃﷯﷯ – θ ﷐𝑑𝑦﷮𝑑𝜃 ﷯ = ﷐𝑑﷮𝑑𝜃﷯﷐﷐4﷐sin﷮𝜃﷯﷮2 + ﷐cos﷮𝜃﷯﷯ −𝜃﷯ ﷐𝑑𝑦﷮𝑑𝜃 ﷯ = ﷐𝑑﷮𝑑𝜃﷯﷐﷐4﷐sin﷮𝜃﷯﷮2 +﷐cos﷮𝜃﷯﷯ ﷯− ﷐𝑑𝜃﷮𝑑𝜃﷯ ﷐𝑑𝑦﷮𝑑𝜃 ﷯ = ﷐𝑑﷮𝑑𝜃﷯﷐﷐4﷐sin﷮𝜃﷯﷮2 +﷐cos﷮𝜃﷯﷯ ﷯− 1 ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐﷐﷐4﷐sin﷮𝜃﷯﷯﷮′﷯﷐2 + ﷐cos﷮𝜃﷯﷯−﷐﷐2 + ﷐cos﷮𝜃﷯﷯﷮′﷯﷐4﷐sin﷮𝜃﷯﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ −1 ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐4﷐cos﷮𝜃﷯ ﷐2 + ﷐cos﷮𝜃﷯﷯−﷐0− ﷐sin﷮𝜃﷯﷯﷐4﷐sin﷮𝜃﷯﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ −1 ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐8﷐cos﷮𝜃 + 4 ﷐𝑐𝑜𝑠2𝜃 + 𝑠𝑖𝑛2𝜃﷯﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ −1 ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐8﷐cos﷮𝜃 + 4﷐1﷯﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ −1 ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐8﷐cos﷮𝜃 + 4﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ −1 ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐8﷐cos﷮𝜃 + 4﷯ − ﷐﷐2 + ﷐cos﷮𝜃﷯﷯2﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐8﷐cos﷮𝜃 + 4﷯ − ﷐4 + 𝑐𝑜𝑠2 𝜃 + 4 𝑐𝑜𝑠𝜃﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐8﷐cos﷮𝜃 + 4﷯ − 4 − 𝑐𝑜𝑠2𝜃 − 4 𝑐𝑜𝑠𝜃﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐8﷐cos﷮𝜃 − 4﷐cos﷮𝜃﷯ + 4 − 4 − 𝑐𝑜𝑠2𝜃﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐4 cos 𝜃 + 0 − 𝑐𝑜𝑠2𝜃﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ ﷐𝑑𝑦﷮𝑑𝑥﷯ = ﷐4﷐cos﷮𝜃 − 𝑐𝑜𝑠2 𝜃﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ For the function to be increasing ﷐𝑑𝑦﷮𝑑𝜃﷯ = ﷐﷐cos﷮𝜃﷯﷐4 − ﷐cos﷮𝜃﷯﷯﷮﷐2 + ﷐cos﷮𝜃﷯﷯2﷯ > 0 Since denominator is a square, We need to show that cos θ ﷐4−﷐cos﷮𝜃﷯﷯ > 0 for θ ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ Now, 0 ≤ cos θ ≤ 1 Multiplying by –1 –1 ≤ –cos θ ≤ 0 Adding 4 both sides –1 + 4 ≤ – cos θ +4 < 0 + 4 3 ≤ 4 – cos2 θ < 4 Thus, (4 – cos2 θ ) is positive And cos θ is also positive Hence cos θ (4 – cos2 θ) is also positive ⇒ cos θ (4 – cos2 θ) > 0 for θ ∈ ﷐0 , ﷐𝜋﷮2﷯ ﷯ Hence 𝑦 = ﷐4﷐sin﷮𝜃﷯﷮2 + 𝑐𝑜𝑠𝜃 ﷯ – θ is an increasing function for θ ∈ ﷐𝟎 , ﷐𝝅﷮𝟐﷯﷯

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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