How do you find the local max and min for #f(x) = x^3  27x#?
Maximum at
Find the critical values of
A critical value
Find
#f(x)=x^327x#
#f'(x)=3x^227#
#3c^227=0#
#3c^2=27#
#c^2=9#
#c=+3#
We know that two critical values, at which maxima or minima could occur, are
We can use either or the first or second derivative test to determine if these are minima or maxima.
First Derivative Test
Examine the change in the function surrounding the critical values.
#f'(4)=21larr"increasing"#
#f'(3)=0#
#f'(2)=15larr"decreasing"#
Since the derivative changes from increasing to decreasing when
#f'(2)=15larr"decreasing"#
#f'(3)=0#
#f'(4)=21larr"increasing"#
Since the derivative changes from decreasing to increasing when
Second Derivative Test
Examine the concavity at each point to determine whether a minimum or maximum should occur.
First, find
#f'(x)=3x^227#
#f''(x)=6x#
Now, find the concavity at each of the critical values.
#f''(3)=18#
Since
#f''(3)=18#
Since
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To find the local maxima and minima for ( f(x) = x^3  27x ), you need to follow these steps:
 Compute the derivative of ( f(x) ), denoted as ( f'(x) ).
 Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points.
 Determine the nature of each critical point using either the first or second derivative test.
 Identify the local maxima and minima based on the nature of the critical points.
Let's go through these steps:

Compute the derivative of ( f(x) ): [ f'(x) = 3x^2  27 ]

Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points: [ 3x^2  27 = 0 ] [ x^2  9 = 0 ] [ x^2 = 9 ] [ x = \pm 3 ]

Determine the nature of each critical point:
 At ( x = 3 ), ( f''(3) = 6(3) = 18 < 0 ), so it's a local maximum.
 At ( x = 3 ), ( f''(3) = 6(3) = 18 > 0 ), so it's a local minimum.

Identify the local maxima and minima:
 Local maximum at ( x = 3 ).
 Local minimum at ( x = 3 ).
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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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