# What are the critical values, if any, of #f(x)= x^(3/4) - 2x^(1/4)#?

By the definition I am accustomed to, they are

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To find the critical values of ( f(x) = x^{3/4} - 2x^{1/4} ), we first need to find the derivative of the function, set it equal to zero, and solve for ( x ).

( f'(x) = \frac{3}{4}x^{-1/4} - \frac{1}{2}x^{-3/4} )

Setting the derivative equal to zero:

( \frac{3}{4}x^{-1/4} - \frac{1}{2}x^{-3/4} = 0 )

Solving for ( x ):

( \frac{3}{4}x^{-1/4} = \frac{1}{2}x^{-3/4} )

( \frac{3}{2} = 2x^{-3/4} )

( \frac{3}{4} = x^{-3/4} )

( x^{-3/4} = \frac{3}{4} )

( x = \left(\frac{3}{4}\right)^{-4/3} )

( x = \left(\frac{4}{3}\right)^{4/3} )

So, the critical value is ( x = \left(\frac{4}{3}\right)^{4/3} ).

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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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