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

Through the chain rule:

Then:

Factoring from the final parentheses:

Rewriting:

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To find the critical points of ( f(x) = \sqrt{e^{\sqrt{x}} - \sqrt{x}} ), we first need to find its derivative, set it equal to zero, and solve for ( x ).

( f'(x) = \frac{1}{2\sqrt{e^{\sqrt{x}} - \sqrt{x}}} \cdot \left(\frac{1}{2\sqrt{x}}\cdot e^{\sqrt{x}} - \frac{1}{2\sqrt{x}}\right) )

Setting ( f'(x) = 0 ), we get:

( \frac{1}{2\sqrt{e^{\sqrt{x}} - \sqrt{x}}} \cdot \left(\frac{1}{2\sqrt{x}}\cdot e^{\sqrt{x}} - \frac{1}{2\sqrt{x}}\right) = 0 )

Since the first factor cannot be zero, we focus on the second factor:

( \frac{1}{2\sqrt{x}}\cdot e^{\sqrt{x}} - \frac{1}{2\sqrt{x}} = 0 )

Solving this equation yields:

( e^{\sqrt{x}} - 1 = 0 )

( e^{\sqrt{x}} = 1 )

( \sqrt{x} = 0 )

( x = 0 )

So, the only critical point of ( f(x) ) is ( x = 0 ).

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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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