Is #f(x)=(3x^3+x^2-2x+7)/(2x-2)# increasing or decreasing at #x=3#?

To determine whether the function ( f(x) = \frac{{3x^3 + x^2 - 2x + 7}}{{2x - 2}} ) is increasing or decreasing at ( x = 3 ), we need to evaluate the sign of its derivative at that point. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing.

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

[ f'(x) = \frac{{(2x - 2)(9x^2 + 2x - 2) - (3x^3 + x^2 - 2x + 7)(2)}}{{(2x - 2)^2}} ]

Now, evaluate ( f'(3) ) to determine if it's positive or negative:

[ f'(3) = \frac{{(2(3) - 2)(9(3)^2 + 2(3) - 2) - (3(3)^3 + (3)^2 - 2(3) + 7)(2)}}{{(2(3) - 2)^2}} ] [ = \frac{{(6 - 2)(81 + 6 - 2) - (81 + 9 - 6 + 7)(2)}}{{(6 - 2)^2}} ] [ = \frac{{(4)(85) - (91)(2)}}{{4}} ] [ = \frac{{340 - 182}}{{4}} ] [ = \frac{{158}}{{4}} ] [ = 39.5 ]

Since ( f'(3) = 39.5 ), which is positive, the function is increasing at ( x = 3 ).