Question

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

Answer 1

At `x=1, f(x)` is decreasing

Explanation:

Let's calculate (by the rule of derivatove of a division):

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

Then let's substitute x=1:

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

`=(-16+6)/4=-5/2<0`

Since `f'(1)<0` then f(x) is decreasing at x=1