Question

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

Answer 1

Increasing at `x_0=-1`

Explanation:

`f(x)=(4x^3+2x^2-2x-3)/(x-2)`

`D_f=(-oo,2)uu(2,+oo)`

`f'(x)=((4x^3+2x^2-2x-3)'(x-2)-(4x^3+2x^2-2x-3)(x-2)')/(x-2)^2`

`=` `((12x^2+4x-2)(x-2)-(4x^3+2x^2-2x-3))/(x-2)^2` `=`

`(12x^3-24x^2+4x^2-8x-2x+4-4x^3-2x^2+2x+3)/(x-2)^2`

`=` `(8x^3-22x^2-8x+7)/(x-2)^2` `=`

`f'(-1)=(-8-22+8+7)/9=-15/9=-5/3<0`

so `f` is decreasing at `x_0=-1` as seen in the graph.