What are the critical values, if any, of #f(x)=x^(4/5) (x − 3)^2#?
They are
Differentiate using the product rule:
graph{y=x^(4/5)(x-3)^2 [-3.01, 6.857, -0.442, 4.49]}
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To find the critical values of ( f(x) = x^{\frac{4}{5}} (x - 3)^2 ), we need to first find the derivative of ( f(x) ), set it equal to zero, and solve for ( x ).
Taking the derivative of ( f(x) ) using the product rule:
( f'(x) = \frac{4}{5}x^{-\frac{1}{5}}(x-3)^2 + x^{\frac{4}{5}}(2(x-3)) )
Setting ( f'(x) ) equal to zero and solving for ( x ):
( \frac{4}{5}x^{-\frac{1}{5}}(x-3)^2 + x^{\frac{4}{5}}(2(x-3)) = 0 )
( \frac{4}{5}x^{-\frac{1}{5}}(x-3)^2 = -2x^{\frac{4}{5}}(x-3) )
( \frac{4}{5}(x-3) = -2x^{\frac{4}{5}} )
( 4(x-3) = -\frac{10}{5}x^{\frac{4}{5}} )
( 4x - 12 = -2x^{\frac{4}{5}} )
( 2x^{\frac{4}{5}} + 4x - 12 = 0 )
This equation can be solved numerically to find the critical values.
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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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