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

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

 

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

  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

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
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.