Example 9 - Chapter 6 Class 12 Application of Derivatives

Last updated at Dec. 16, 2024 by Teachoo

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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) (0 , 2π) = (0 , π) ∪ (π , 2π) From 1st part f(𝑥) is strictly decreasing in (0 , π) And from 2nd part f(𝑥) is strictly increasing in (π , 2π) Thus, f(𝒙) is neither increasing nor decreasing in (0, 2π)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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