Ex 6.2,17 - Chapter 6 Class 12 Application of Derivatives

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Ex 6.2,17 Prove that the function f given by f (𝑥) = log cos 𝑥 is strictly decreasing on ﷐0,﷐𝜋﷮2﷯﷯ and strictly increasing on﷐﷐𝜋﷮2﷯,𝜋﷯ f﷐𝑥﷯ = log cos 𝑥 We need to show f﷐𝑥﷯ is strictly decreasing on ﷐0 , ﷐𝜋﷮2﷯﷯ & Strictly increasing on ﷐﷐𝜋﷮2﷯ , 𝜋﷯ i.e. We need to show f’﷐𝑥﷯ < 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ & f’ ﷐𝑥﷯ > 0 for 𝑥 ∈ ﷐﷐𝜋﷮2﷯ , 𝜋﷯ Finding f’﷐𝑥﷯ f﷐𝑥﷯=log cos 𝑥 f’﷐𝑥﷯ = ﷐1﷮﷐cos﷮𝑥﷯﷯ . ﷐𝑑﷮𝑑𝑥﷯ ﷐﷐cos﷮𝑥﷯﷯ f’﷐𝑥﷯ = ﷐1﷮﷐cos﷮𝑥﷯﷯ × ﷐−﷐sin﷮𝑥﷯﷯ f’ ﷐𝑥﷯ = ﷐−﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯﷯ Case 1: For 0 < 𝒙 < ﷐𝝅﷮𝟐﷯ We know that cos 𝑥 > 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ & sin 𝑥 > 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ f’﷐𝑥﷯ =﷐−﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯﷯ = ﷐(−)﷐+﷯﷮﷐+﷯﷯ for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ < 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ Hence f’﷐𝑥﷯ < 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ Thus f﷐𝑥﷯ is strictly decreasing for 𝒙 ∈ ﷐𝟎 , ﷐𝝅﷮𝟐﷯﷯ Case 2: For ﷐𝝅﷮𝟐﷯ < 𝒙 < π We know that cos 𝑥 < 0 for 𝑥 ∈ ﷐﷐𝜋﷮2﷯ , 𝜋﷯ & sin 𝑥 > 0 for 𝑥 ∈ ﷐﷐𝜋﷮2﷯ , 𝜋﷯ Now, f’﷐𝑥﷯ =﷐−﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯﷯= ﷐﷐−﷯(+)﷮﷐−﷯﷯ > 0 ⇒ f’﷐𝑥﷯ > 0 for 𝑥 ∈ ﷐﷐𝜋﷮2﷯ , 𝜋﷯ ∴ f’(x) is strictly increasing for 𝒙 ∈ ﷐﷐𝝅﷮𝟐﷯ , 𝝅﷯