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

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  1. Chapter 6 Class 12 Application of Derivatives
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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’ (π‘₯) = – sin π‘₯ 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€, 2 Ο€), and f (π‘₯) = cos π‘₯ f’ (π‘₯) = βˆ’ sin π‘₯ Note:- sin π‘₯ < 0 for π‘₯ ∈ (Ο€ , 2Ο€) So, – sin 𝒙 > 0 for π‘₯ ∈ (Ο€ , 2Ο€) β‡’ 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€) & 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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