How do you find the critical numbers of #f(x)=x^(2/3)+x^(1/3)#?
This function has one critical value of
The local minimum value is #f(1/2)=1/2^(2/3)+2^(1/3)=(1+2)/(2^(2/3)) =3/(2^(2/3)) approx 1.89#.
Here's the graph of this function:
graph{x^(2/3)+x^(1/3) [10, 10, 5, 5]}
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To find the critical numbers of ( f(x) = x^\frac{2}{3} + x^\frac{1}{3} ):
 Find the derivative of ( f(x) ) with respect to ( x ).
 Set the derivative equal to zero and solve for ( x ).
 Any ( x ) value obtained in step 2 is a potential critical number. However, you should check each potential critical number to ensure it is not an endpoint of the domain and that it doesn't make the derivative undefined.
Let's proceed with the steps:

Find the derivative of ( f(x) ): [ f'(x) = \frac{2}{3}x^{\frac{1}{3}}  \frac{1}{3}x^{\frac{4}{3}} ]

Set the derivative equal to zero: [ \frac{2}{3}x^{\frac{1}{3}}  \frac{1}{3}x^{\frac{4}{3}} = 0 ]

Solve for ( x ): [ \frac{2}{3}x^{\frac{1}{3}} = \frac{1}{3}x^{\frac{4}{3}} ] [ 2x^{1} = x^{4} ] [ 2 = x^{3} ] [ x = \sqrt[3]{2} ]

Check the critical number ( x = \sqrt[3]{2} ):
 Ensure it's within the domain of ( f(x) ): Since ( x ) can be any real number except 0, ( x = \sqrt[3]{2} ) satisfies the domain.
 Check if it makes the derivative undefined: Since ( x = \sqrt[3]{2} ) is not 0, it does not make the derivative undefined.
Therefore, the only critical number of ( f(x) = x^\frac{2}{3} + x^\frac{1}{3} ) is ( x = \sqrt[3]{2} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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