Consider the function $f\left(x\right)=x^3{e}^{-2x}$. Its critical points are located at $(0,0)$ and $(\frac{3}{2},\frac{27}{8e^3})$. Use the Second Derivative Test to identify whether these points are local maxima, minima, or neither.

Neither maximum nor minimum at $(0,0)$,

Local maximum at $(\frac{3}{2},\frac{27}{8e^3})$.

Local minimum at $(0,0)$,

Local maximum at $(\frac{3}{2},\frac{27}{8e^3})$.

Local maximum at $(0,0)$,

Local minimum at $(\frac{3}{2},\frac{27}{8e^3})$.

Neither maximum nor minimum at $(0,0)$,

Local minimum at $(\frac{3}{2},\frac{27}{8e^3})$.