How do you find the critical points for #f(x)=x^3-2x^2+3x#?

Answer 1

#x=1/3(2-isqrt5) and x=1/3(2+isqrt5)#

To find the critical points you first take the derivative using the power:

#d/dx=x^n=nx^(n-1)#
The derivative of #f(x)=x^3-2x^2+3x# is:
#d/dx=3x^2-4x+3#

Now you set it equal to zero and solve:

#3x^2-4x+3=0#

When you solve it, it will yield complex roots:

#x=1/3(2-isqrt5) and x=1/3(2+isqrt5)#

These two solutions are your critical numbers.

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Answer 2

To find the critical points of ( f(x) = x^3 - 2x^2 + 3x ), you need to find the values of ( x ) where the derivative ( f'(x) ) is zero or undefined.

  1. Calculate the derivative ( f'(x) ) of the function ( f(x) ): [ f'(x) = \frac{d}{dx} (x^3 - 2x^2 + 3x) ] [ f'(x) = 3x^2 - 4x + 3 ]

  2. Set ( f'(x) ) to zero and solve for ( x ) to find the critical points: [ 3x^2 - 4x + 3 = 0 ]

This quadratic equation doesn't factor easily, so you can use the quadratic formula: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ] where ( a = 3 ), ( b = -4 ), and ( c = 3 ).

[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(3)}}{2(3)} ] [ x = \frac{4 \pm \sqrt{16 - 36}}{6} ] [ x = \frac{4 \pm \sqrt{-20}}{6} ] [ x = \frac{4 \pm 2i\sqrt{5}}{6} ]

The critical points for ( f(x) = x^3 - 2x^2 + 3x ) are the values of ( x ) where ( f'(x) = 0 ), which are: [ x = \frac{4 + 2i\sqrt{5}}{6} ] [ x = \frac{4 - 2i\sqrt{5}}{6} ]

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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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