# What are the critical points of #f(x) =x^(e^x)#?

There are none.

We will use logarithmic differentiation, which is helpful when finding a derivative such as this one, where we have a function whose power is another function. To use this method, begin by taking the natural logarithm of both sides of the equation.

Now take the derivative of both sides of the equation. The left side will need the chain rule, since we're dealing with a composite function (a function within a function). On the right side, we'll use the product rule.

We have three parts:

The second is never true.

So, we have found no critical points.

graph{x^(e^x) [-.2, 3, -20, 100]}

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To find the critical points of ( f(x) = x^{e^x} ), we need to find where its derivative equals zero or does not exist.

[ f'(x) = (e^x \cdot x^{e^x - 1}) + (x^{e^x} \cdot e^x) \ln(x) ]

Setting ( f'(x) ) equal to zero, we get:

[ (e^x \cdot x^{e^x - 1}) + (x^{e^x} \cdot e^x) \ln(x) = 0 ]

[ e^x \cdot x^{e^x - 1} + x^{e^x} \cdot e^x \ln(x) = 0 ]

[ x^{e^x - 1}(e^x + e^x \ln(x)) = 0 ]

Critical points occur where ( x = 0 ) or where ( e^x + e^x \ln(x) = 0 ). However, ( x = 0 ) is not a critical point because it is not in the domain of the function. Therefore, we need to solve ( e^x + e^x \ln(x) = 0 ).

This equation does not have an algebraic solution, so we can find approximate values of critical points using numerical methods or graphical methods.

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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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