Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12   1. Chapter 6 Class 12 Application of Derivatives
2. Serial order wise
3. Examples

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Example 9 Prove that the function given by f (x) = cos x is (a) strictly decreasing in (0, π) f(𝑥) = cos 𝑥 f’(𝑥) = – sin 𝑥 Since, sin 𝑥 > 0 for 𝑥 ∈ (0, π) So, –sin 𝒙 < 0 for 𝑥 ∈ (0, π) ∴ f (𝑥) < 0 for 𝑥 ∈ (0 , π) So, f is Strictly decreasing in (0 , π) Example 9 Prove that the function given by f (x) = cos x is (b) strictly increasing in (π, 2π), and f (𝑥) = cos 𝑥 f’(𝑥) = − sin 𝑥 Since sin 𝑥 < 0 for 𝑥 ∈ (π , 2π) So, – sin 𝒙 > 0 for 𝑥 ∈ (π , 2π) ∴ f’(𝑥) > 0 for 𝑥 ∈ (π , 2π) So, f is strictly increasing in (π , 2π) Rough sin π = 0 sin 5𝜋/4 = sin ("π + " 𝜋/4) = −sin (𝜋/4) = (−1)/√2 sin 2π = sin (π + π ) = − sin π = 0 Value of sin 𝑥 < 0 for 𝑥 ∈ (π , 2π) Example 9 Prove that the function given by f (x) = cos x is (c) neither increasing nor decreasing in (0, 2π). (0 , 2π) = (0 , π) ∪ (π , 2π) From 1st part f (𝑥) is strictly decreasing in (0 , π) & from 2nd part f (𝑥) is strictly increasing in (π , 2π) Thus, f (𝒙) is neither increasing nor decreasing in (0, 2π)

Examples 