What are the absolute extrema of # f(x)= xln(3x) in [1,e]#?
Find the roots of first derivative you get
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To find the absolute extrema of ( f(x) = x  \ln(3x) ) on the interval ([1, e]), we first find the critical points by setting the derivative equal to zero and then evaluate ( f(x) ) at these critical points and at the endpoints of the interval.

Find the derivative of ( f(x) ): [ f'(x) = 1  \frac{1}{x} ]

Set the derivative equal to zero and solve for ( x ) to find critical points: [ 1  \frac{1}{x} = 0 ] [ \frac{1}{x} = 1 ] [ x = 1 ]

Evaluate ( f(x) ) at the critical point and endpoints: [ f(1) = 1  \ln(3) \approx 1  1.0986 \approx 0.0986 ] [ f(e) = e  \ln(3e) = e  (\ln(3) + \ln(e)) = e  (\ln(3) + 1) \approx 1.5849 ]

Compare the values of ( f(x) ) at the critical point and endpoints:
 ( f(1) \approx 0.0986 )
 ( f(e) \approx 1.5849 )
Therefore, the absolute minimum of ( f(x) ) on the interval ([1, e]) is approximately ( 0.0986 ) at ( x = 1 ), and the absolute maximum is approximately ( 1.5849 ) at ( x = e ).
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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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