# Is #f(x)=(4x^3+2x^2-2x-3)/(x-2)# increasing or decreasing at #x=-1#?

Increasing at

so

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To determine if ( f(x) = \frac{{4x^3 + 2x^2 - 2x - 3}}{{x - 2}} ) is increasing or decreasing at ( x = -1 ), we can use the first derivative test.

First, find the derivative of ( f(x) ): [ f'(x) = \frac{{d}}{{dx}}\left( \frac{{4x^3 + 2x^2 - 2x - 3}}{{x - 2}} \right) ] [ f'(x) = \frac{{(12x^2 + 4x - 2)(x - 2) - (4x^3 + 2x^2 - 2x - 3)(1)}}{{(x - 2)^2}} ] [ f'(x) = \frac{{12x^3 - 4x^2 - 24x^2 + 8x - 2x + 4 - 4x^3 - 2x^2 + 2x + 3}}{{(x - 2)^2}} ] [ f'(x) = \frac{{-6x^2 - 16x + 7}}{{(x - 2)^2}} ]

Now, plug in ( x = -1 ) into ( f'(x) ) to determine the sign of the derivative at ( x = -1 ): [ f'(-1) = \frac{{-6(-1)^2 - 16(-1) + 7}}{{(-1 - 2)^2}} ] [ f'(-1) = \frac{{-6 + 16 + 7}}{{(-3)^2}} ] [ f'(-1) = \frac{{17}}{{9}} ]

Since ( f'(-1) > 0 ), ( f(x) ) is increasing at ( 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.

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