Question

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

Answer 1

First we have to calculate `f'(x)`.

`f'(x)=((2x-3)*(x^2+1)-(x^2-3x-2)*2x)/(x^2+1)^2`

`f'(x)=(2x^3-3x^2+2x-3-2x^3+6x^2+4x)/(x^2+1)^2`

`f'(x)=(3x^2+6x-3)/(x^2+1)^2`

To find if `f(x)` is increasing or decreasing in `x=3` we have to calculate the value of `f'(3)`

`f'(3)=(3*9+6*3-3)/(9+1)^2=42/100>0`

`f'(3)>0` so `f(x)` is increasing in `x=3`