Locate the critical points of $h\left(x\right)=x^3-6x^2+9x$, and use the Second Derivative Test to identify whether these points are local maxima, minima, or neither.

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.

Locate the critical points of $f\left(x\right)=x^4\ln \left(x\right)-4x^4$, and use the Second Derivative Test to identify whether these points are local maxima, minima, or neither.