Question

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

Answer 1

f(x) is increasing at x=2

Explanation:

First, let's differentiate f(x):

`f'(x)=((1-e^x-xe^x)(x-1)-(x-xe^x))/(x-1)^2`

`=(cancelx-1cancel(-xe^x)-e^x-x^2e^xcancel(+xe^x)cancel-x+xe^x)/(x-1)^2`

`=(-1-e^x-x^2e^x+xe^x)/(x-1)^2`

Then, let's calculate

`f'(2)=(-1-e^2-4e^2+2e^2)/9`

`=(-1-3e^2)/9<0`

Then f(x) is increasing at x=2

{ graph{(x-xe^x)/(x-1) [-20, 20, -227.2, 124.8]}