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Question
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Solution
\[f\left( x \right) = \cos x\]
\[f'\left( x \right) = - \sin x\]
\[\left( i \right) \] \[\text { Here },\]
\[0 < x < \pi\]
\[ \Rightarrow \sin x > 0 \left[ \because \text { Sine function is positive in first and second quadrant } \right]\]
\[ \Rightarrow - \sin x < 0\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in \left( 0, \pi \right)\]
\[\text { So, f(x) is strictly decreasing on } \left( 0, \pi \right) . \]
\[\left( ii \right) \] \[\text { Here, }\]
\[\pi < x < 2\pi\]
\[ \Rightarrow \sin x < 0 \left[ \because \text { Sine function is negative in third and fourth quadrant} \right]\]
\[ \Rightarrow - \sin x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \pi, 2\pi \right)\]
\[\text { So,f(x)is strictly increasing on } \left( \pi, 2\pi \right) . \]
\[\left( iii \right) \] \[\text { From eqs. (1) and (2), we get }\]
\[f(x)\text { is strictly decreasing on } \left( 0, \pi \right) \text { and is strictly increasing on } \left( \pi, 2\pi \right) . \]
\[\text { So,}f\left( x \right) \text { is neither increasing nor decreasing on}\left( 0, 2\pi \right).\]
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