How do you find the exact relative maximum and minimum of the polynomial function of #p(x) =80+108xx^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:

Find the derivative of ( p(x) ): ( p'(x) = 108  3x^2 ).

Set ( p'(x) ) equal to zero to find the critical points: ( 108  3x^2 = 0 ). Solving for ( x ): ( x^2 = 36 ). ( x = \pm 6 ).

Find the second derivative: ( p''(x) = 6x ).

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 onesided 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 onesided 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 onesided 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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