Advertising spend of maximum revenue

Statement

The revenue of an advertising campaign, as a function of the spend $x$ (in thousands of €), is $R(x)=100x\,e^{-0{.}1x}$ (in thousands of €). Find the spend that maximizes revenue and the maximum revenue.

Solution

$R'(x)=100e^{-0{.}1x}+100x\,(-0{.}1)e^{-0{.}1x}=100e^{-0{.}1x}(1-0{.}1x)=0\Rightarrow x=10$. For $x<10$ $R'>0$, for $x>10$ $R'<0$: maximum at $x=10$. $R(10)=1000\,e^{-1}=\frac{1000}{e}\approx\boxed{367{.}88}\ \text{(thousands of €)}.$