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

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Ex 6.2,3 Find the intervals in which the function f given by f (𝑥) = Sin 𝑥 is (a) strictly increasing in ﷐0 , ﷐𝜋﷮2﷯﷯ f (𝑥) = sin 𝑥 f’(𝑥) = cos 𝑥 Since cos 𝑥 > 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ So, f’ (𝑥) > 0 for 𝑥 ∈ ﷐0 , ﷐𝜋﷮2﷯﷯ Thus, f is strictly increasing in ﷐0 , ﷐𝜋﷮2﷯﷯ Ex 6.2,3 Find the intervals in which the function f given by f (𝑥) = Sin x is (b) strictly decreasing ﷐﷐𝜋﷮2﷯,𝜋﷯ f (𝑥) = sin 𝑥 f’(𝑥) = cos 𝑥 Since cos 𝑥 < 0 for 𝑥 ∈ ﷐﷐𝜋﷮2﷯ , 𝜋﷯ So, f’ (𝑥) < 0 for 𝑥 ∈ ﷐﷐𝜋﷮2﷯ , π﷯ Thus, f is strictly decreasing in ﷐﷐𝜋﷮2﷯ π﷯ Ex 6.2,3 Find the intervals in which the function f given by f (𝑥) = Sin x is (c)neither increasing nor decreasing in (0 ,π ) Step 1: f (𝑥) = sin 𝑥 f’ (𝑥) = cos 𝑥 Step 2: Putting f’(𝑥) = 0 cos x= 0 𝑥 = ﷐𝜋﷮2﷯ Step 3: Since 𝑥 ∈ (0 , π) we start number line from 0 to π The point 𝑥= ﷐𝜋﷮2﷯ divides the interval (0 , 𝜋) into two disjoint interval i.e. ﷐0 , ﷐𝜋﷮2﷯﷯ , ﷐﷐𝜋﷮2﷯ 𝜋﷯ From 1st part , f’(𝑥) is strictly increasing in ﷐0 ﷐𝜋﷮2﷯﷯ & from 2nd part, f(𝑥) is strictly decreasing in ﷐﷐𝜋﷮2﷯ 𝜋﷯ Hence f(𝑥) is neither increasing nor decreasing in (0 , π)