How do use the first derivative test to determine the local extrema #f(x) = x³+3x²9x+15#?
Now, we can use the graph of the first derivative to determine whether those points are extrema and which type of extrema:
graph{3(x+3)(x1) [10, 10, 5.21, 5.21]}
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To use the first derivative test to determine the local extrema of ( f(x) = x^3 + 3x^2  9x + 15 ):
 Find the first derivative of the function ( f(x) ).
 Set the derivative equal to zero and solve for ( x ). These are the critical points.
 Test the intervals around each critical point by checking the sign of the derivative.
 If the sign changes from positive to negative at a critical point, it indicates a local maximum.
 If the sign changes from negative to positive at a critical point, it indicates a local minimum.
 If the sign does not change, there is no local extremum at that point.
Let's go through these steps:

Find the first derivative: ( f'(x) = 3x^2 + 6x  9 ).

Set ( f'(x) ) equal to zero and solve for ( x ): ( 3x^2 + 6x  9 = 0 ). Factor or use the quadratic formula to find the solutions.

Once you have the critical points, test the intervals around each critical point by plugging in values into ( f'(x) ) to determine the sign.
After applying the first derivative test, you can determine the local extrema of the function ( f(x) ).
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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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