Let $f\left(x\right)=x^3-6x^2+12x-3\text{.}$ Then at $x=2,f\left(x\right)$ has 

We have,

$f\left(x\right)=x^3-6x^2+12x-3$

$\Rightarrow f^{\mathrm{\prime }}\left(x\right)=3x^2-12x+12,f^{\mathrm{\prime }}\left(x\right)=6x-12$ and $f^{\prime \prime \mathrm{\prime }}\left(x\right)=6.$

We observe that $f^{\mathrm{\prime }}\left(2\right)=0,f^{\prime \prime }\left(2\right)=0$ but $f^{\prime \prime \mathrm{\prime }}\left(2\right)\ne 0$.

So, $x=2$ is neither a point of local maximum nor a point of local minimum. In fact, it is a point of inflexion.