How do use the first derivative test to determine the local extrema #f(x) = x³+3x²-9x+15#?

Answer 1

#x=-3# is a local maximum and #x=1# a local minimum.

First we look for points that might be the extrema by solving #f'(x)=0#: #f'(x)=3x^2+6x-9=3(x+3)(x-1)# #3(x+3)(x-1)=0 => x=-3 vv x=1# #x=-3# and #x=1# are two potential extrema.

Now, we can use the graph of the first derivative to determine whether those points are extrema and which type of extrema:

if when passing through point #x# the first derivative changes it's sign from positive to negative then #x# is a local maximum and if from negative to positive - a locac minimum.

graph{3(x+3)(x-1) [-10, 10, -5.21, 5.21]}

In this case #x=-3# is a local maximum and #x=1# a local minimum.
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Answer 2

To use the first derivative test to determine the local extrema of ( f(x) = x^3 + 3x^2 - 9x + 15 ):

  1. Find the first derivative of the function ( f(x) ).
  2. Set the derivative equal to zero and solve for ( x ). These are the critical points.
  3. 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:

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

  2. 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.

  3. 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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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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