Question

Is `f(x)=(-x^3+x^2+2x-11)/(x-1)` increasing or decreasing at `x=2`?

Answer 1

Gradient is positive hence function is increasing at `x=2`

Explanation:

Find `f'(x)` for increasing/decreasing.
Use quotient rule since the function is a fraction.

Let `f=-x^3+x^2+2x-11` and `g=x-1`
`f'=-3x^2+2x+2`
`g'=1`
`f'(x)=(f'g-g'f)/(g^2)`
`f'(x)=((-3x^2+2x+2)(x-1)-(1)(-x^3+x^2+2x-11))/(x-1)^2`
`f'(x)=(-2x^3+4x^2-2x+9)/(x-1)^2`

`f'(2)=(-2(2)^3+4(2)^2-2(2)+9)/((2)-1)^2`
`=(-16+16-4+9)/1`
`=5`

Gradient is positive hence function is increasing at `x=2`