What are the critical values, if any, of #f(x) = x^3 + x^2 - x #?
Critical points:
This cannot be factored and solved. We need to use the quadratic equation:
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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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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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