Question

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

Answer 1

The function is increasing at `x = -1`

Explanation:

Differentiate the function to get the gradient. If the gradient is positive the function is increasing, and if it is negative the function is decreasing.
`f(x) = (1-x)/(x+2)^3`
`f'(x) = (-x(x+2)^3 -(1-x)(3(x+2)^2))/(x+2)^6`
`=(-x(x+2)^3 - 3(x+2)^2 + 3x(x+2)^2)/(x+2)^6`
`=((x+2)^2(-x(x+2)-3 +3x))/(x+2)^6`
`=(-x^2-2x-3+3x)/(x+2)^4`
`=(-x^2+x+3)/(x+2)^4`
At `x = -1` this becomes
`(-(-1)^2 +(-1) +3)/(-1+2)^4 = (-1-1+3)/(1)^4`
`=1`

This is positive therefore the function is increasing.