# Is #f(x)=(-2x^3+x^2-2x-4)/(4x-2)# increasing or decreasing at #x=0#?

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To determine whether ( f(x) = \frac{-2x^3 + x^2 - 2x - 4}{4x - 2} ) is increasing or decreasing at ( x = 0 ), you can analyze the sign of the derivative of ( f(x) ) at that point.

To find the derivative, apply the quotient rule:

[ f'(x) = \frac{(4x - 2)(-6x^2 + 2x - 2) - (-2x^3 + x^2 - 2x - 4)(4)}{(4x - 2)^2} ]

Now, substitute ( x = 0 ) into ( f'(x) ) to determine whether the derivative is positive (indicating increasing), negative (indicating decreasing), or zero (indicating a possible local extremum) at ( x = 0 ).

[ f'(0) = \frac{(4 \times 0 - 2)(-6 \times 0^2 + 2 \times 0 - 2) - (-2 \times 0^3 + 0^2 - 2 \times 0 - 4)(4)}{(4 \times 0 - 2)^2} ]

[ f'(0) = \frac{(-2)(-2) - (-4)(4)}{(-2)^2} = \frac{4 - (-16)}{4} = \frac{20}{4} = 5 ]

Since ( f'(0) > 0 ), the function is increasing at ( x = 0 ).

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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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