# How do you find the inflection points of the graph of the function: # f(x)=x^(1/3)#?

Examine the behaviour of

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To find the inflection points of the function ( f(x) = x^{1/3} ), follow these steps:

- Calculate the second derivative of the function.
- Set the second derivative equal to zero and solve for ( x ).
- The solutions obtained in step 2 represent the ( x )-coordinates of the inflection points.
- To confirm whether these points are inflection points, check the concavity of the function on either side of each candidate point.

Let's go through the steps:

- ( f(x) = x^{1/3} )
- First derivative: ( f'(x) = \frac{1}{3}x^{-2/3} )
- Second derivative: ( f''(x) = -\frac{2}{9}x^{-5/3} )
- Set ( f''(x) = 0 ): [ -\frac{2}{9}x^{-5/3} = 0 ] [ x^{-5/3} = 0 ] Since ( x^{-5/3} ) cannot be zero for any real value of ( x ), there are no solutions.
- There are no inflection points for the function ( f(x) = x^{1/3} ).

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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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