How do use the first derivative test to determine the local extrema #f(x)= 4x^3  3x^4#?
The local extrema occur at point where the first derivative is equal to zero (critical points) and where the slope (as given by the first derivative) switches between positive and negative
The situation can be further examined by considering the graph below: graph{4x^33x^4 [2.733, 2.744, 1.366, 1.374]}
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To use the first derivative test to determine the local extrema of ( f(x) = 4x^3  3x^4 ):
 Find the first derivative of the function, ( f'(x) ).
 Set ( f'(x) = 0 ) and solve for ( x ) to find critical points.
 Determine the sign of ( f'(x) ) around each critical point.
 If ( f'(x) ) changes sign from positive to negative at a critical point, it indicates a local maximum. If ( f'(x) ) changes sign from negative to positive, it indicates a local minimum.
Let's go through the steps:

( f(x) = 4x^3  3x^4 )
( f'(x) = 12x^2  12x^3 )

Set ( f'(x) = 0 ):
( 12x^2  12x^3 = 0 )
( 12x^2(1  x) = 0 )
( x = 0 ) or ( x = 1 )
These are the critical points.

Test the sign of ( f'(x) ) around each critical point:
Test ( x = 1 ): ( f'(1) = 12(1)^2  12(1)^3 = 12  12 = 0 ) (neither positive nor negative)
Test ( x = 0 ): ( f'(0) = 12(0)^2  12(0)^3 = 0 ) (neither positive nor negative)
Test ( x = 1 ): ( f'(1) = 12(1)^2  12(1)^3 = 12  12 = 0 ) (neither positive nor negative)

Since the sign of ( f'(x) ) doesn't change around the critical points, there are no local extrema for this function.
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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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