How do use the first derivative test to determine the local extrema #f(x)=x^4-4x^3+4x^2+6 #?

Answer 1

You get: 2 minima and 1 maximum.

We can study the derivative of the function:
#f'(x)=4x^3-12x^2+8x#
Set the derivative equal to zero:
#f'(x)=0#
#4x^3-12x^2+8x=0#
#4x(x^2-3x+2)=0#
With solutions:
#x_1=0#
#x^2-3x+2=0# using the Quadratic Formula:
#x_(2,3)=(3+-sqrt(9-8))/2=(3+-1)/2#
two solutions:
#x_2=2#
#x_3=1#
We can now study the sign of the derivative setting: #f'(x)>0# getting:

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Answer 2

To use the first derivative test to determine the local extrema of ( f(x) = x^4 - 4x^3 + 4x^2 + 6 ):

  1. Find the first derivative of the function: ( f'(x) = 4x^3 - 12x^2 + 8x ).
  2. Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points. ( 4x^3 - 12x^2 + 8x = 0 ) Factor out ( 4x ): ( 4x(x^2 - 3x + 2) = 0 ) Factor the quadratic equation: ( 4x(x - 1)(x - 2) = 0 ) So, critical points are ( x = 0, x = 1, ) and ( x = 2 ).
  3. Use the first derivative test to determine the local extrema:
    • Test the interval ( (-\infty, 0) ): Pick a test point ( x = -1 ). ( f'(-1) = 4(-1)^3 - 12(-1)^2 + 8(-1) = -4 - 12 - 8 = -24 < 0 ) Therefore, ( f(x) ) is decreasing on ( (-\infty, 0) ).
    • Test the interval ( (0, 1) ): Pick a test point ( x = 0.5 ). ( f'(0.5) = 4(0.5)^3 - 12(0.5)^2 + 8(0.5) = 0 < 0 ) Therefore, ( f(x) ) is decreasing on ( (0, 1) ).
    • Test the interval ( (1, 2) ): Pick a test point ( x = 1.5 ). ( f'(1.5) = 4(1.5)^3 - 12(1.5)^2 + 8(1.5) = 0 > 0 ) Therefore, ( f(x) ) is increasing on ( (1, 2) ).
    • Test the interval ( (2, \infty) ): Pick a test point ( x = 3 ). ( f'(3) = 4(3)^3 - 12(3)^2 + 8(3) = 0 > 0 ) Therefore, ( f(x) ) is increasing on ( (2, \infty) ).
  4. Based on the first derivative test:
    • ( f(x) ) has a local maximum at ( x = 1 ).
    • ( f(x) ) has a local minimum at ( x = 2 ).
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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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