# What are the local extrema of #f(x)= xe^-x#?

graph{xe^-x [-10, 10, -5, 5]}

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The local extrema of ( f(x) = x e^{-x} ) occur where its derivative equals zero or where the derivative is undefined.

To find the derivative of ( f(x) ), we use the product rule:

( f'(x) = e^{-x} - xe^{-x} )

To find where the derivative equals zero:

( e^{-x} - xe^{-x} = 0 )

Solving this equation gives:

( x = 1 )

Now, to classify whether this point is a local maximum or minimum, we can check the second derivative:

( f''(x) = -e^{-x} + xe^{-x} )

Plugging ( x = 1 ) into the second derivative:

( f''(1) = -e^{-1} + e^{-1} = 0 )

Since the second derivative at ( x = 1 ) is zero, the test fails and we need to use other methods to determine the nature of the extremum. Typically, this involves examining the behavior of the function around ( x = 1 ).

By observing the behavior of the function, we see that it changes from increasing to decreasing around ( x = 1 ), indicating that ( x = 1 ) is a local maximum.

Therefore, the local maximum of ( f(x) = x e^{-x} ) is at ( x = 1 ).

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When evaluating a one-sided 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 one-sided 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 one-sided 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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