Question

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

Answer 1

In `x=1` the function is decreasing.

Explanation:

Evaluate:

`(df)/dx = ( ( x^2+3) (-6x-3) - 2x(-3x^2-3x+2))/(x^2+3)^2`

`(df)/dx = ( cancel(-6x^3)-18x-3x^2 -9 cancel(+6x^3) +6x^2-4x)/(x^2+3)^2`

`(df)/dx = ( 3x^2-22x -9 )/(x^2+3)^2`

For `x=1`:

`[(df)/dx]_(x=1) = ( 3-22 -9 )/(1+3)^2 =-28/16= -7/4 < 0`

Then in `x=1` the function is decreasing.

graph{(-3x^2-3x+2)/(x^2+3) [-1.494, 3.103, -1.49, 0.808]}