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

There are none.

graph{(e^(x^(1/2))+x^(1/2)e^(x^(1/2))-1)/(4x^(3/4)(e^(x^(1/2))-1))-(2x^(1/4)(e^(x^(1/2))-1))/(4x^(3/4)(e^(x^(1/2))-1)) [-10, 10, -5, 5]}

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To find the critical points of ( f(x) = \sqrt{\sqrt{x} e^{\sqrt{x}} - \sqrt{x}} - \sqrt{x} ), we need to first find its derivative and then solve for values of ( x ) where the derivative equals zero or is undefined.

The derivative of ( f(x) ) with respect to ( x ) can be found using the chain rule and product rule:

[ f'(x) = \frac{d}{dx} \left( \sqrt{\sqrt{x} e^{\sqrt{x}} - \sqrt{x}} - \sqrt{x} \right) ]

[ = \frac{1}{2\sqrt{\sqrt{x} e^{\sqrt{x}} - \sqrt{x}}} \cdot \frac{d}{dx} \left( \sqrt{x} e^{\sqrt{x}} - \sqrt{x} \right) - \frac{1}{2\sqrt{x}} ]

[ = \frac{1}{2\sqrt{\sqrt{x} e^{\sqrt{x}} - \sqrt{x}}} \left( \frac{1}{2\sqrt{x}} e^{\sqrt{x}} + \sqrt{x} \cdot \frac{d}{dx} e^{\sqrt{x}} - \frac{1}{2\sqrt{x}} \right) - \frac{1}{2\sqrt{x}} ]

[ = \frac{1}{2\sqrt{\sqrt{x} e^{\sqrt{x}} - \sqrt{x}}} \left( \frac{1}{2\sqrt{x}} e^{\sqrt{x}} + \sqrt{x} \cdot \frac{1}{2\sqrt{x}} e^{\sqrt{x}} - \frac{1}{2\sqrt{x}} \right) - \frac{1}{2\sqrt{x}} ]

[ = \frac{1}{4(\sqrt{x} e^{\sqrt{x}} - \sqrt{x})} \left( \frac{e^{\sqrt{x}} + 1}{\sqrt{x}} - 1 \right) - \frac{1}{2\sqrt{x}} ]

[ = \frac{e^{\sqrt{x}} + 1 - 4\sqrt{x}}{4\sqrt{x}(\sqrt{x} e^{\sqrt{x}} - \sqrt{x})} - \frac{1}{2\sqrt{x}} ]

Now, we need to find the critical points by setting the derivative equal to zero:

[ f'(x) = 0 ]

[ \frac{e^{\sqrt{x}} + 1 - 4\sqrt{x}}{4\sqrt{x}(\sqrt{x} e^{\sqrt{x}} - \sqrt{x})} - \frac{1}{2\sqrt{x}} = 0 ]

Solving this equation for ( x ) will give us the critical points. However, this equation is complex and doesn't have a simple solution. We may need to use numerical methods to approximate the critical points.

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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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