How do you find the critical numbers of #f(x)=x^(2/3)+x^(-1/3)#?

Answer 1

This function has one critical value of #x#, at #x=1/2#, and gives a local minimum value there of #f(1/2)=3/(2^(2/3)) approx 1.89#

The derivative of this function is #f'(x)=2/3 x^(-1/3)-1/3 x^(-4/3)#. Getting a common denominator of #3x^{4/3}# allows us to write #f'(x)=(2x-1)/(3x^{4/3})#. This is equal to zero when #2x-1=0#, which means #x=1/2#. It's also undefined at #x=0#, though the original function is undefined there as well.
Hence, the value #x=1/2# is the only critical value of #f#. The sign of #f'# changes from negative to positive as #x# increases through #1/2#, implying that there is a local (relative) minimum point at #x=1/2#.

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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Answer 2

To find the critical numbers of ( f(x) = x^\frac{2}{3} + x^\frac{-1}{3} ):

  1. Find the derivative of ( f(x) ) with respect to ( x ).
  2. Set the derivative equal to zero and solve for ( x ).
  3. 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:

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

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

  3. 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} ]

  4. 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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Answer from HIX Tutor

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.

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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