What are the critical values, if any, of #f(x) = x^3 + x^2 - x #?

Answer 1

Critical points: #(0.43, -0.49)# and #(-0.77, -2.23)#

We are given: #f(x) = x^3 +x^2-x#
We compute the derivative: #f'(x) = 3x^2+x-1#
We set the derivative to zero to find critical points: #f'(x) = 0# #3x^2+x-1 = 0#

This cannot be factored and solved. We need to use the quadratic equation:

#x = (-b += sqrt(b^2-4ac))/(2a)# where #a = 3#, #b=1#, and #c=-1#.
This gives: #x = 0.43# and #x = -0.77#
Substituting these values into #f(x)# gives: #f(0.43) = -0.49# #f(-0.77) = -2.23#
Hence the critical points are: #(0.43, -0.49)# and #(-0.77, -2.23)#
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Answer 2

To find the critical values of ( f(x) = x^3 + x^2 - x ), we first find its derivative:

[ f'(x) = 3x^2 + 2x - 1 ]

Then, we set the derivative equal to zero and solve for ( x ) to find the critical points:

[ 3x^2 + 2x - 1 = 0 ]

Using the quadratic formula, we find:

[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ]

where ( a = 3 ), ( b = 2 ), and ( c = -1 ). Plugging in these values, we get:

[ x = \frac{{-2 \pm \sqrt{{2^2 - 4(3)(-1)}}}}{{2(3)}} ] [ x = \frac{{-2 \pm \sqrt{{4 + 12}}}}{{6}} ] [ x = \frac{{-2 \pm \sqrt{{16}}}}{{6}} ] [ x = \frac{{-2 \pm 4}}{{6}} ]

So the critical values are:

[ x_1 = \frac{{-2 + 4}}{{6}} = \frac{1}{3} ] [ x_2 = \frac{{-2 - 4}}{{6}} = -1 ]

Therefore, the critical values of ( f(x) = x^3 + x^2 - x ) are ( x = \frac{1}{3} ) and ( x = -1 ).

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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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