How do you find the local maximum and minimum values of #f ' (x) = (x^2 -9)#?

Answer 1
at local max or min #f'(x)# is zero.
=> #x^2 -9 = 0# => #x^2 = 9# => # x = +-sqrt9 = +- 3# so at #x =+- 3# , the function is either max or min (locally).
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Answer 2

To find the local maximum and minimum values of ( f'(x) = x^2 - 9 ), follow these steps:

  1. Find the critical points by setting ( f'(x) ) equal to zero and solving for ( x ).
  2. 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:

  1. 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.
  2. 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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Answer from HIX Tutor

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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