How do you find the local maximum and minimum values of #f ' (x) = (x^2 9)#?
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To find the local maximum and minimum values of ( f'(x) = x^2  9 ), follow these steps:
 Find the critical points by setting ( f'(x) ) equal to zero and solving for ( x ).
 Determine the nature of each critical point (whether it corresponds to a local maximum, minimum, or neither) using the first or second derivative test.
First, find the critical points by solving ( f'(x) = 0 ):
[ x^2  9 = 0 ] [ x^2 = 9 ] [ x = \pm 3 ]
Next, use the first or second derivative test to determine the nature of each critical point:

First derivative test:
 ( f'(3) = (3)^2  9 = 0 )
 ( f'(3) = (3)^2  9 = 0 ) Since the first derivative changes sign from negative to positive at ( x = 3 ) and from positive to negative at ( x = 3 ), ( x = 3 ) corresponds to a local minimum and ( x = 3 ) corresponds to a local maximum.

Second derivative test (optional but can be used for confirmation):
 Find the second derivative: ( f''(x) = 2x )
 Evaluate the second derivative at each critical point:
 ( f''(3) = 2(3) = 6 ) (negative, concave down, local minimum)
 ( f''(3) = 2(3) = 6 ) (positive, concave up, local maximum)
Therefore, the local minimum occurs at ( x = 3 ) with ( f(3) = (3)^2  9 = 0 ), and the local maximum occurs at ( x = 3 ) with ( f(3) = (3)^2  9 = 0 ).
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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.
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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