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

You get: 2 minima and 1 maximum.

We can study the derivative of the function:

Set the derivative equal to zero:

With solutions:

two solutions:

We can now study the sign of the derivative setting:

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

- Find the first derivative of the function: ( f'(x) = 4x^3 - 12x^2 + 8x ).
- 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 ).
- 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) ).

- 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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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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