Ex 6.2,3 - Chapter 6 Class 12 Application of Derivatives

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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) ∴ f’(𝑥) < 0 for 𝑥 ∈ (0 , π) Thus, f is strictly increasing in ("0 , " 𝜋/2) Rough cos 0 = 1 cos 𝜋/4 = 1/√2 cos 𝜋/2 = 0 Value of cos⁡𝑥 > 0 for (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) ∴ f’(𝑥) < 0 for 𝑥 ∈ (𝜋/2 " , π" ) Thus, f is strictly decreasing in (𝜋/2 " π" ) Rough cos 𝜋/2 = 0 cos 3𝜋/4 = co𝑠 ("π − " 𝜋/4) = – cos⁡𝜋/4 = (−1 )/√2 Value of cos 𝑥 < o for 𝑥 ∈ (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)(0 , π) = (0 , 𝜋/2) ∪ (𝜋/2,𝜋) From 1st part f(𝑥) is strictly increasing in (0 , 𝜋/2) And from 2nd part f(𝑥) is strictly decreasing in (𝜋/2,𝜋) Thus, f(𝒙) is neither increasing nor decreasing in (0, π)