For 0<x<180 verify the monotony of the function xcosx+sin(-x).

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We have the function f(x) = x*cos x + sin (-x)

f(x) = x*cos x - sin x

The first derivative of f(x) is f'(x)

=> f'(x) = cos x - x*(sin x) - cos x

=> f'(x) = -x*sin x

When 0< x< 180 we have sin x as...

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We have the function f(x) = x*cos x + sin (-x)

f(x) = x*cos x - sin x

The first derivative of f(x) is f'(x)

=> f'(x) = cos x - x*(sin x) - cos x

=> f'(x) = -x*sin x

When 0< x< 180 we have sin x as positive, therefore the first derivative -x*sin x is negative, so the function is decreasing.

Therefore the function is decreasing for the given range of x.

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