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Question

Show that f(x)=cosx is an decreasing function on (0,π) increasing in (π,0) and neither increasing nor decreasing in (π,π).

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Solution

f(x)=cosx
f(x)=sinx

(i) Now 0<x<π
sinx<0
f(x)<0
so f(x) is decreasing on (0,π)

(ii) π<x<0
sinx>0
f(x)>0
so f(x) is increasing on (π,0)

(iii) π<x<π
In this interval
sinx>0 as well as sinx<0
Hence f(x) is neither increasing nor decreasing in [π,π].

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