# What are the critical points of #f(x)=x^(2/3) *e^ (x^2-4)#?

there is only one critical point, at x = 0

critical points are the points in a function where the slope changes from positive to negative or vice versa, therefore these points have potential to be the maximum or minimum point of a function.

this change of slop can happen at two types of situations, where the derivative is 0 and where it is undefined these places can also be the maximum or minimum value.

therefore the only critical point is where x = 0 you can see this on the graph of the function graph{x^(2/3) * e^(x^2 - 4) [-11.25, 11.25, -5.625, 5.625]}

the only place where the derivative or the slope changes from negative to positive s at x = 0

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To find the critical points of ( f(x) = x^{\frac{2}{3}} \cdot e^{x^2 - 4} ), we first need to find its derivative, set it equal to zero, and then solve for ( x ).

[ f'(x) = \frac{2}{3}x^{-\frac{1}{3}} \cdot e^{x^2 - 4} + 2x \cdot e^{x^2 - 4} ]

Setting ( f'(x) ) equal to zero:

[ \frac{2}{3}x^{-\frac{1}{3}} \cdot e^{x^2 - 4} + 2x \cdot e^{x^2 - 4} = 0 ]

We can't solve this equation explicitly for ( x ) in terms of elementary functions, but we can find the critical points by setting each factor equal to zero and solving for ( x ).

For the first factor: [ \frac{2}{3}x^{-\frac{1}{3}} \cdot e^{x^2 - 4} = 0 ] [ x^{-\frac{1}{3}} = 0 ]

This has no real solution because ( x ) cannot be zero.

For the second factor: [ 2x \cdot e^{x^2 - 4} = 0 ] [ 2x = 0 ] [ x = 0 ]

So, the only critical point 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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