If #f(x)=sinxxcosx#, how the function behaves in the intervals #(0,pi)# and #(pi,2pi)# i.e. whether it is increasing or decreasing?
Between
between
graph{5.54, 14.46, 6.36, 3.64]}
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To determine if ( f(x) = \sin(x)  x\cos(x) ) is increasing or decreasing in the intervals ((0,\pi)) and ((\pi, 2\pi)), we need to examine the derivative of ( f(x) ) in each interval.

Interval ((0,\pi)):
 Take the derivative of ( f(x) ): ( f'(x) = \cos(x)  \cos(x) + x\sin(x) = x\sin(x) ).
 Since ( \sin(x) ) is positive in ((0,\pi)) and ( x ) is also positive in this interval, ( f'(x) ) is positive.
 Therefore, ( f(x) ) is increasing in ((0,\pi)).

Interval ((\pi, 2\pi)):
 Again, take the derivative of ( f(x) ): ( f'(x) = x\sin(x) ).
 In this interval, ( \sin(x) ) is negative, but ( x ) is positive.
 Therefore, ( f'(x) ) is negative.
 Consequently, ( f(x) ) is decreasing in ((\pi, 2\pi)).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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