# Is #f(x)=(-2x^3+9x^2-5x+6)/(x-2)# increasing or decreasing at #x=0#?

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To determine whether the function ( f(x) = \frac{{-2x^3 + 9x^2 - 5x + 6}}{{x - 2}} ) is increasing or decreasing at ( x = 0 ), we can analyze its derivative at that point.

- First, find the derivative of the function ( f(x) ).
- Evaluate the derivative at ( x = 0 ).
- Determine the sign of the derivative at ( x = 0 ).

The function is increasing at ( x = 0 ) if the derivative is positive, and it's decreasing if the derivative is negative.

Let's find the derivative:

( f(x) = \frac{{-2x^3 + 9x^2 - 5x + 6}}{{x - 2}} )

Using the quotient rule to differentiate, we get:

( f'(x) = \frac{{(x - 2)(-6x^2 + 18x - 5) - (-2x^3 + 9x^2 - 5x + 6)(1)}}{{(x - 2)^2}} )

Now, evaluate ( f'(0) ) to determine whether the function is increasing or decreasing at ( x = 0 ).

( f'(0) = \frac{{(-2)(-5) - (6)(1)}}{{(-2)^2}} = \frac{{10 - 6}}{{4}} = \frac{4}{4} = 1 )

Since ( f'(0) = 1 > 0 ), the function ( f(x) ) 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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