What are the local extrema of #f(x)= x^3-x+3/x#?

Answer 1

#x_1= -1# is a maximum
#x_2= 1# is a minimum

First find the critical points by equating the first derivative to zero:

#f'(x) = 3x^2-1-3/x^2#
#3x^2-1-3/x^2 = 0#
As #x!=0# we can multiply by #x^2#
#3x^4-x^2-3=0#
#x^2=frac(1+-sqrt(1+24)) 6 #
so #x^2=1# as the other root is negative, and #x=+-1#

Then we look at the sign of the second derivative:

#f''(x) = 6x+6/x^3#
#f''(-1) = -12 <0#
#f''(1) = 12>0#

so that:

#x_1= -1# is a maximum #x_2= 1# is a minimum

graph{x^3-x+3/x [-20, 20, -10, 10]}

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Answer 2

To find the local extrema of ( f(x) = x^3 - x + \frac{3}{x} ), we first find the derivative ( f'(x) ) and then solve for critical points by setting ( f'(x) = 0 ).

[ f'(x) = 3x^2 - 1 - \frac{3}{x^2} ]

Setting ( f'(x) = 0 ):

[ 3x^2 - 1 - \frac{3}{x^2} = 0 ]

[ 3x^4 - x^2 - 3 = 0 ]

This is a quadratic equation in terms of ( x^2 ). Let ( u = x^2 ):

[ 3u^2 - u - 3 = 0 ]

Solving this quadratic equation for ( u ), we get:

[ u = \frac{1 \pm \sqrt{1 + 4(3)(3)}}{6} ]

[ u = \frac{1 \pm \sqrt{37}}{6} ]

Since ( u = x^2 ), take the square root to find ( x ):

[ x = \pm \sqrt{\frac{1 \pm \sqrt{37}}{6}} ]

Now, we evaluate the second derivative ( f''(x) ) to determine the nature of the critical points:

[ f''(x) = 6x + \frac{6}{x^3} ]

Evaluate ( f''(x) ) at the critical points to determine the concavity:

  1. Substitute ( x = -\sqrt{\frac{1 + \sqrt{37}}{6}} ) into ( f''(x) ) to determine concavity.
  2. Substitute ( x = \sqrt{\frac{1 + \sqrt{37}}{6}} ) into ( f''(x) ) to determine concavity.
  3. Substitute ( x = -\sqrt{\frac{1 - \sqrt{37}}{6}} ) into ( f''(x) ) to determine concavity.
  4. Substitute ( x = \sqrt{\frac{1 - \sqrt{37}}{6}} ) into ( f''(x) ) to determine concavity.

The local extrema occur at the points where the concavity changes (from concave up to concave down or vice versa) or at the endpoints of the interval if applicable. These points need to be evaluated to determine if they are local maxima or minima.

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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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