How do you find the exact relative maximum and minimum of the polynomial function of #p(x) =80+108x-x^3 #?
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To find the relative maximum and minimum of the polynomial function ( p(x) = 80 + 108x - x^3 ):
- Find the critical points by setting the derivative equal to zero and solving for ( x ).
- Determine the nature of each critical point by evaluating the second derivative at each critical point.
- Identify the relative maximum and minimum based on the behavior of the function around each critical point.
Let's follow these steps:
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Find the derivative of ( p(x) ): ( p'(x) = 108 - 3x^2 ).
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Set ( p'(x) ) equal to zero to find the critical points: ( 108 - 3x^2 = 0 ). Solving for ( x ): ( x^2 = 36 ). ( x = \pm 6 ).
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Find the second derivative: ( p''(x) = -6x ).
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Evaluate the second derivative at each critical point: ( p''(6) = -36 ) (negative, indicating a relative maximum). ( p''(-6) = 36 ) (positive, indicating a relative minimum).
Therefore, the relative maximum occurs at ( x = 6 ) and the relative minimum occurs at ( x = -6 ).
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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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