I would like your help to determine whether there is a fault in my reasoning.
Let's say that we have the following shape in $\mathbb{R}^3$:
$B=\\{(x,y,z) \in \mathbb{R}^3 | x^4+y^4+z^4 \le 1\\}$
Some unknown function $f:\mathbb{R}^3\rightarrow\mathbb{R}$, that is defined everywhere and is $C^3$, has a maximum point in $B$, off the axes.
I define the following Lagrange multiplier problem, for every $0\lt R\lt 1$:
$\begin{cases} \text{ext.}& f(x,y,z) \\\\ \text{s.t.}& B_R(x,y,z)=x^4+y^4+z^4-R=0 \end{cases}$
The solution of such problem is $Df(x,y,z)=\lambda DB_R(x,y,z)$
Therefore $\frac{\partial f}{\partial x}=\lambda\cdot 4x^3$, etc.
If we derive a second time, we get $\frac{\partial^2 f}{\partial x^2}=12\lambda x^2, \frac{\partial^2 f}{\partial x\partial y}=0$, etc.
So the Hessian matrix for a max point $u$ of $f$ in $B$, which is in radius $R$, is defined because it is an inner point, and is supposedly known as: $Hess(f)(u)=diag(12\lambda x^2,12\lambda y^2,12\lambda z^2)$
The Hessian is positive definite, so $u$ cannot be a maximum point.
The only possibility left is that the maximum point is received on $\partial B$ (the surface of B), where a Hessian is not defined over B.
Is this a valid usage of Lagrange multipliers?