Question

Find the points of maximum and minimum and the intervals of monotonicity of the following function:
$f\left(x\right)=x\cdot e^{-3x}$

Answer 1

$f\left(x\right)=x.e^{-3x}$

$f^{\prime }\left(x\right)=e^{-3x}-3x.e^{-3x}=0$

$e^{-3x}\ne 0butx=\frac{1}{3}$

$f^{\prime \prime }\left(x\right)=-3e^{-3x}-3e^{-3x}+9x{e}^{-3x}\Rightarrow f^{\prime \prime }\left(\frac{1}{3}\right)=-3e^{-1}<0$

The function has absolute minimum at $x=\frac{1}{3}$ and decreases continously from there ,hence monotonously from $x=\frac{1}{3}to\infty$