To find out if a function #f(x)# is increasing or decreasing at a given point #x_0# you can calculate the first derivative #f'(x_0)#.
If the derivative #f'(x_0)# is greater than zero, then the function is increasing, if it is negative, then the function is decreasing.
Here the derivative can be calculated using the quotient rule:
#f(x)=(2x^3-7x^2+3x+1)/(x-3)#
#f'(x)=((2x^3-7x^2+3x+1)'(x-3)-(2x^3-7x^2+3x+1)(x-3)')/(x-3)^2#
#f'(x)=((6x^2-14x+3)(x-3)-(2x^3-7x^2+3x+1))/(x-3)^2#
#f'(x)=(6x^3-32x^2+45x-9-2x^3+7x^2-3x-1)/(x-3)^2#
#f'(x)=(4x^3-25x^2+42x-10)/(x-3)^2#
Now we calculate the value of #f'(2)#
#f'(2)=(4*2^3-25*2^2+42*2-10)/(2-3)^2#
#f'(2)=32-100+84-10=6#
#f'(2)# is greater than zero, so #f(x)# is increasing at #x=2#
