How do you find the critical numbers of # f(x)=x^(1/5)-x^(-4/5)#?

Answer 1

The only critical number is #x=-4#

The critical numbers of a function are the points where its derivative equals zero, which means they are the solutions of the equation:

#(df)/dx = 0#
#1/5x^(-4/5)+4/5x^(-9/5) = 0#
#1/5x^(-4/5)(1+4/x) = 0#
As #x^(-4/5) !=0# for any value of #x# the only solutions are given by:
#1+4/x = 0#
#4/x = -1#
#x=-4#
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Answer 2

To find the critical numbers of ( f(x) = x^{1/5} - x^{-4/5} ), you first need to find the derivative of the function ( f'(x) ). Then, set the derivative equal to zero and solve for ( x ). The values of ( x ) obtained from solving this equation will be the critical numbers of the function.

Here's the process:

  1. Find the derivative ( f'(x) ) using the power rule and the chain rule. [ f'(x) = \frac{1}{5}x^{-4/5} + \frac{4}{5}x^{-9/5} ]

  2. Set the derivative equal to zero and solve for ( x ). [ \frac{1}{5}x^{-4/5} + \frac{4}{5}x^{-9/5} = 0 ]

  3. Solve the equation for ( x ). This equation simplifies to: [ x^{-4/5}(1 + 4x^{-5/5}) = 0 ] [ x^{-4/5}(1 + 4x^{-1}) = 0 ]

The critical numbers will be the solutions for ( x ) in this equation.

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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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