Curve study/e^(-2x)-2e^(-x)/x not negative/Morse potential/Exercise

Discuss the behavior of the function graph of

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=e^{-2x}-2e^{-x}.$

Determine especially the monotonicity behavior, the extrema of ${}f$, ${}\operatorname {lim} _{x\rightarrow \infty }\,f(x)$ and also for the derivative ${}f'$.

Create a solution