Example 12 - Find intervals where f(x) = sin 3x is decreasing - Examples

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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Example 12 Find intervals in which the function given by f (x) = sin 3x, x, ∈ ﷐0, ﷐𝜋﷮2﷯﷯ is (a) increasing (b) decreasing. f﷐𝑥﷯ = sin 3𝑥 where 𝑥 ∈ ﷐0 ,﷐𝜋﷮2﷯﷯ Step 1 :- Finding f’(x) f﷐𝑥﷯ = sin 3𝑥 f’﷐𝑥﷯ = ﷐𝑑﷐﷐sin﷮3𝑥﷯﷯﷮𝑑𝑥﷯ f’﷐𝑥﷯ = cos 3𝑥 . ﷐𝑑﷐3𝑥﷯﷮𝑑𝑥﷯= cos 3𝑥. 3 = 3. cos 3𝑥 Step 2: Putting f’﷐𝑥﷯ = 0 3 cos 3𝑥 = 0 cos 3𝑥 = 0 We know that cos θ = 0 When θ = ﷐𝜋﷮2﷯ & ﷐3𝜋﷮2﷯ ⇒ 3𝑥 = ﷐𝜋﷮2﷯ & 3𝑥 = ﷐3𝜋﷮2﷯ 𝑥 = ﷐𝜋﷮2 ×3﷯ & 𝑥 = ﷐3𝜋﷮2 × 3﷯ 𝑥 = ﷐𝜋﷮6﷯ & 𝑥 = ﷐𝜋﷮2﷯ Since 𝑥 = ﷐𝜋﷮6﷯ ∈ ﷐0 ,﷐𝜋﷮2﷯﷯ & 𝑥 = ﷐𝜋﷮2﷯ ∈ ﷐0,﷐𝜋﷮2﷯﷯ both values of 𝑥 are valid Step 3: Plotting point Since 𝑥 ∈ ﷐0 ,﷐𝜋﷮2﷯ ﷯ we start number line from 0 & end at ﷐𝜋﷮2﷯ Point 𝑥 = ﷐𝜋﷮6﷯ divide the interval ﷐0 ,﷐𝜋﷮2﷯﷯ into two disjoint intervals ﷐0 ,﷐𝜋﷮6﷯﷯ and ﷐﷐𝜋﷮6﷯, ﷐𝜋﷮2﷯﷯ Step 4: Checking sign of f’﷐𝑥﷯ f’﷐𝑥﷯ = 3. cos 3𝑥 Case 1 In 𝑥 ∈ ﷐0 ,﷐𝜋﷮6﷯﷯ 0<𝑥<﷐𝜋﷮6﷯ 3×0<3𝑥<﷐3𝜋﷮6﷯ 0<3𝑥<﷐𝜋﷮2﷯ So when 𝑥 ∈ ﷐0 ,﷐𝜋﷮6﷯﷯, then 3𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ And we know that cos 𝜃>0 for 𝜃 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ cos 3x >0 for 3x ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ cos 3x >0 for x ∈ ﷐0 , ﷐𝜋﷮6﷯﷯ 3 cos 3x >0 for x ∈ ﷐0 , ﷐𝜋﷮6﷯﷯ 𝑓′(𝑥)>0 for x ∈ ﷐0 , ﷐𝜋﷮6﷯﷯ Since f’(x) ≥ 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮6﷯﷯ Thus, f(x) is increasing for 𝑥 ∈ ﷐0 , ﷐𝜋﷮6﷯﷯ Case 2 Since 𝑥 ∈ ﷐﷐𝜋﷮6﷯, ﷐𝜋﷮2﷯﷯ ﷐𝜋﷮6﷯<𝑥<﷐𝜋﷮2﷯ 3× ﷐𝜋﷮6﷯<3𝑥<﷐3𝜋﷮2﷯ ﷐𝜋﷮2﷯<3𝑥<﷐3𝜋﷮2﷯ So when 𝑥 ∈﷐﷐𝜋﷮6﷯ , ﷐𝜋﷮2﷯﷯, then 3𝑥 ∈ ﷐﷐𝜋﷮2﷯ , ﷐3𝜋﷮2﷯﷯ We know that, cos 𝜃<0 for 𝜃 ∈ ﷐﷐𝜋﷮2﷯ , ﷐3𝜋﷮2﷯﷯ cos 3𝑥<0 for 3𝑥 ∈ ﷐﷐𝜋﷮2﷯ , ﷐3𝜋﷮2﷯﷯ cos 3𝑥<0 for 𝑥 ∈ ﷐﷐𝜋﷮6﷯ , ﷐𝜋﷮2﷯﷯ 3 cos 3𝑥<0 for 𝑥 ∈ ﷐﷐𝜋﷮6﷯ , ﷐𝜋﷮2﷯﷯ f‘(x) <0 for 𝑥 ∈ ﷐﷐𝜋﷮6﷯ , ﷐𝜋﷮2﷯﷯ Since f’(x) ≤ 0 for 𝑥 ∈ ﷐﷐𝜋﷮6﷯,﷐𝜋﷮2﷯﷯ Thus, f(x) is decreasing for 𝑥 ∈ ﷐﷐𝜋﷮6﷯,﷐𝜋﷮2﷯﷯ Thus, f(x) is increasing for 𝒙 ∈ ﷐𝟎 , ﷐𝝅﷮𝟔﷯﷯ & f(x) is strictly decreasing for 𝒙 ∈ ﷐﷐𝝅﷮𝟔﷯ , ﷐𝝅﷮𝟐﷯﷯

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