How do use the first derivative test to determine the local extrema #F(x) = 2x^3  9x^2 + 24x + 40#?
Determine the critical points of the function and check to see if the first derivative changes sign around these points.
The first derivative test allows you to determine whether or not a critical point of a function is also a local minimum or a local maximum.
This can be done by checking to see if the first derivative of the function changes signs around these critical points.
So, start by determining the first derivative for your function
This is equivalent to
Since you have no domain restrictions for this function, both solutions will be critical points.
Now check to see if the first derivative changes sign around these points. Since you have two critical points, you're going to need 3 intervals.
and
Therefore, your function has
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To use the first derivative test to determine the local extrema of ( F(x) = 2x^3  9x^2 + 24x + 40 ):

Find the first derivative of ( F(x) ): ( F'(x) = 6x^2  18x + 24 ).

Set ( F'(x) ) equal to zero and solve for ( x ) to find critical points:
( 6x^2  18x + 24 = 0 ). 
Solve the quadratic equation to find the critical points:
( x^2 + 3x  4 = 0 )
( (x + 4)(x  1) = 0 )
( x = 4, , x = 1 ) 
Determine the sign of ( F'(x) ) in the intervals created by the critical points (∞, 4), (4, 1), and (1, ∞).
For ( x = 4 ): ( F'(5) = 6(5)^2  18(5) + 24 = 54 > 0 )
So, ( F'(x) ) is positive to the left of ( x = 4 ).For ( x = 1 ): ( F'(0) = 6(0)^2  18(0) + 24 = 24 > 0 )
So, ( F'(x) ) is positive between ( x = 4 ) and ( x = 1 ).For ( x = 2 ): ( F'(2) = 6(2)^2  18(2) + 24 = 12 < 0 )
So, ( F'(x) ) is negative to the right of ( x = 1 ). 
Based on the signs of ( F'(x) ), determine the nature of the local extrema:
 At ( x = 4 ), there is a local maximum because ( F'(x) ) changes from positive to negative.
 At ( x = 1 ), there is a local minimum because ( F'(x) ) changes from negative to positive.
Therefore, the function ( F(x) ) has a local maximum at ( x = 4 ) and a local minimum at ( x = 1 ).
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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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