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

increasing at x = 0

We evaluate f'(a) to find out if a function is increasing or decreasing at x = a.

• At x = a, f(x) is increasing if f'(a) > 0.

F(x) is increasing at x = 0 because f'(0) > 0. The graph {(4x^3-2x^2-x-3)/(x-2) [-10, 10, -5, 5]} shows this.

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

First, find the derivative of ( f(x) ) using the quotient rule:

[ f'(x) = \frac{{(x - 2)(12x^2 - 4x - 1) - (4x^3 - 2x^2 - x - 3)(1)}}{{(x - 2)^2}} ]

Now, evaluate ( f'(0) ) to determine if the function is increasing or decreasing at ( x = 0 ). If ( f'(0) > 0 ), then the function is increasing at ( x = 0 ); if ( f'(0) < 0 ), then the function is decreasing at ( x = 0 ).

[ f'(0) = \frac{{-2(-1) - (-3)(1)}}{{(-2)^2}} = \frac{{2 + 3}}{{4}} = \frac{5}{4} ]

Since ( f'(0) > 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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- How do you find the intervals of increasing and decreasing using the first derivative given #y=x(x^2-3)#?
- How do you use Rolle's Theorem on a given function #f(x)#, assuming that #f(x)# is not a polynomial?

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