I have a following problem and I am not sure if I understand correctly how to classify stationary points.
The function is given by:
\begin{equation}
f(a, b, c) = a^2b + b^2c + c^2a,
\end{equation}
hence the first order conditions are:
\begin{equation}\label{first}
\frac{\partial f}{\partial x} = 2ab + c^2 = 0
\end{equation}
\begin{equation}\label{second}
\frac{\partial f}{\partial y} = a^2 +2bc = 0
\end{equation}
\begin{equation}\label{third}
\frac{\partial f}{\partial z} = b^2 +2ac = 0
\end{equation}
With one unique solution $a=b=c=0$.
Now so far I think I understand things, but I have now problem with classifying the stationary point. In a 2 variable case I would simply calculate second order derivatives and then the determinant of hessian at a stationary point.
\begin{equation}
\begin{aligned}
H(a,b,c) & =
\begin{bmatrix}
2b & 2a & 2c\\
2a &2c& 2b\\
2c&2b&2a
\end{bmatrix}
\end{aligned}
\end{equation}
now I dont know if this is correct but just kinda trying to extend the two variable case I would calculate the following at the stationary point:
\begin{equation}
2b2c2a – |H|
\end{equation}
where $|H|$ is the determinant of hessian.
At a stationary point I would have
\begin{equation}
0 – 0\geq 0
\end{equation}
So this should not be a saddle point since the above equation is not negative, but also since the second order derivatives are exactly zero at the point it could be both convex or concave – I am completely lost at this point…
Best Answer
The function $f(a,b,c)$ does not have relative minimum nor relative maximum nor saddle point at $(0,0,0)$ The eigenvalues for $H_f(0,0,0) = 0_{3\times 3}$ are all null.
Attached a plot showing the maniford $f(a,b,c)=0$