How do use the first derivative test to determine the local extrema #y=x(sqrt(8-x^2))#?
graph{xsqrt(8-x^2) [-5, 5, -6, 6]}
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To use the first derivative test to determine the local extrema of ( y = x\sqrt{8 - x^2} ):
- Find the first derivative of the function ( y ) with respect to ( x ).
- Set the derivative equal to zero and solve for ( x ).
- Determine the critical points.
- Use the first derivative test to analyze the behavior of the function around each critical point.
- If the derivative changes sign from positive to negative at a critical point, it indicates a local maximum.
- If the derivative changes sign from negative to positive at a critical point, it indicates a local minimum.
- Identify any local extrema based on the results of the first derivative test.
Let's proceed with these steps:
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Find the first derivative of ( y ) with respect to ( x ): [ y = x\sqrt{8 - x^2} ] [ y' = \sqrt{8 - x^2} + x\left(\frac{1}{2\sqrt{8 - x^2}}\right)(-2x) ] [ y' = \sqrt{8 - x^2} - \frac{x^2}{\sqrt{8 - x^2}} ]
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Set the derivative equal to zero and solve for ( x ): [ \sqrt{8 - x^2} - \frac{x^2}{\sqrt{8 - x^2}} = 0 ] [ \sqrt{8 - x^2} = \frac{x^2}{\sqrt{8 - x^2}} ] [ (8 - x^2) = x^2 ] [ 8 = 2x^2 ] [ x^2 = 4 ] [ x = \pm 2 ]
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Determine the critical points: ( x = 2 ) and ( x = -2 ).
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Use the first derivative test:
- For ( x = 2 ): Evaluate ( y' ) for a value slightly less than 2 and a value slightly more than 2 to determine the sign change.
- For ( x = -2 ): Evaluate ( y' ) for a value slightly less than -2 and a value slightly more than -2 to determine the sign change.
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Identify any local extrema based on the results of the first derivative test.
Alternatively, you can directly analyze the behavior of the function in the interval ((-∞, -2)), ((-2, 2)), and ((2, ∞)).
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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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