Question

Find all intervals on which the function f(x)=(-x+2)e^-x is increasing. If there is no such interval, submit an empty answer.

Answer 1

We have function $f(x) = (-x+2)*e^{-x}$ and we must find interval, where it is increasing. Firstly, let's find the derivative of function:

$f'(x) = -e^{-x} + (-x)(-1)e^{-x} - 2e^{-x}=-3e^{-x} + xe^{-x} =$

$= e^{-x} (x-3)$

We know that function $e^{-x}$ is always positive function and descending function, so $e^{-x} >0$ for $\forall x\in R$

For increase function f(x) we must find inteval, where $f'(x) \ge0$

$e^{-x} (x-3) \ge 0 \implies x-3 \ge 0 \implies x \ge 3$

So we have answer $x\in[3; +\infty)$, where function is increasing