What are the local extrema of #f(x)= xlnxxe^x#?
This function has no local extrema.
Note that
It may be instructive to look at this graphically:
graph{xlog(x)xe^x [0.105, 1, 1.175, 0.075]}
graph{1+log(x)(x+1)*e^x [0.105, 1, 3, 0.075]}
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To find the local extrema of ( f(x) = x \ln(x)  xe^x ), we need to find the critical points by setting the derivative equal to zero and then checking the second derivative to determine whether each critical point corresponds to a local minimum, local maximum, or neither.

Find the first derivative: [ f'(x) = \ln(x) + 1  e^x  xe^x ]

Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points.

Once you have critical points, evaluate the second derivative at each critical point to determine the concavity and thus the nature of the extrema.

If ( f''(x) > 0 ), it's a local minimum. If ( f''(x) < 0 ), it's a local maximum. If ( f''(x) = 0 ), the test is inconclusive.

Any critical points that yield a change in sign of the second derivative indicate a local extremum.

Additionally, check the behavior of the function at the boundaries of the domain (if any) to ensure no additional extrema are missed.
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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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