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

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

The graph{(-x^3+x^2-3x-4)/(4x-2) [-20, 20, -10, 10]} shows that f(x) is increasing at x = 0 since f'(0) > 0.

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

Taking the derivative of ( f(x) ) with respect to ( x ) yields:

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

Now, substituting ( x = 0 ) into ( f'(x) ) gives:

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

Since ( f'(0) = -\frac{1}{2} ), which is negative, ( f(x) ) is decreasing at ( x = 0 ).

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