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Example 9 - Prove that f(x) = cos x is (a) strictly decreasing

Example 9 - Chapter 6 Class 12 Application of Derivatives - Part 2

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Example 9 - Chapter 6 Class 12 Application of Derivatives - Part 3

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Example 9 Prove that the function given by f (x) = cos x is (a) strictly decreasing in (0, Ο€) f(π‘₯) = cos π‘₯ f’(𝒙) = – sin 𝒙 Since, sin π‘₯ > 0 for π‘₯ ∈ (0, Ο€) So, –sin 𝒙 < 0 for π‘₯ ∈ (0, Ο€) ∴ f’(π‘₯) < 0 for π‘₯ ∈ (0 , Ο€) So, f is strictly decreasing in (0 , Ο€) Example 9 Prove that the function given by f (x) = cos x is (b) strictly increasing in (Ο€, 2Ο€), and f (π‘₯) = cos π‘₯ f’(𝒙) = βˆ’ sin 𝒙 Since sin π‘₯ < 0 for π‘₯ ∈ (Ο€ , 2Ο€) So, – sin 𝒙 > 0 for π‘₯ ∈ (Ο€ , 2Ο€) ∴ f’(π‘₯) > 0 for π‘₯ ∈ (Ο€ , 2Ο€) So, f is strictly increasing in (Ο€ , 2Ο€) Example 9 Prove that the function given by f (x) = cos x is (c) neither increasing nor decreasing in (0, 2Ο€).(0 , 2Ο€) = (0 , Ο€) βˆͺ (Ο€ , 2Ο€) From 1st part f(π‘₯) is strictly decreasing in (0 , Ο€) And from 2nd part f(π‘₯) is strictly increasing in (Ο€ , 2Ο€) Thus, f(𝒙) is neither increasing nor decreasing in (0, 2Ο€)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.