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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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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€) (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

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.