How do you find the critical numbers of # f(x)=x^(1/5)-x^(-4/5)#?
The only critical number is
The critical numbers of a function are the points where its derivative equals zero, which means they are the solutions of the equation:
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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:
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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} ]
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Set the derivative equal to zero and solve for ( x ). [ \frac{1}{5}x^{-4/5} + \frac{4}{5}x^{-9/5} = 0 ]
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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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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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