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Question -

Show that f(x) =cos x is a decreasing function on (0, π), increasing in (–π, 0) and neitherincreasing nor decreasing in (–π, π).



Answer -

Given f(x) = cos x

⇒ 

⇒ f’(x)= –sin x

Taking different region from 0 to 2π

Let x ∈ (0,π).

⇒ Sin(x)> 0

⇒ –sinx < 0

⇒ f’(x)< 0

Thus f(x) is decreasing in (0, π)

Let x ∈ (–π,o).

⇒ Sin(x) < 0

⇒ –sinx > 0

⇒ f’(x)> 0

Thus f(x) is increasing in (–π, 0).

Therefore, from above condition we find that

⇒ f(x) is decreasing in (0, π) and increasing in (–π, 0).

Hence, condition for f(x) neither increasing nor decreasing in(–π, π)

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