Multivariable Function Optimization with Semidefinite Hessian Matrix

Question

I'm writing to ask for support in carrying out an exercise about the optimization of a multivariable function. The function is: $f(x,y,z)=x^2+y^2z+z^2-2x$
Clearly the domain is $\mathbb{R}^3$, and $f$ is infinitely differentiable on it.
Well... I started to calculate critical points. It turned out to be only one of them, which is: $P=(1,0,0)$. Then I determined the Hessian Matrix: $Hf(P=(1,0,0))=\begin{pmatrix}2 & 0 & 0 \\ 0 &0 &0 \\ 0 & 0 & 2 \end{pmatrix}$
It's associated to a positive semidefinite quadratic form, so the point $P=(1,0,0)$ must be a local minimum or a saddle point, but we can't conclude anything more. I thought to calculate the sign of the increment $\Delta f=f(x,y,z)-f(P)=x^2+y^2z+z^2-2x-(-1)=(x-1)^2+y^2z+z^2$, in a neighbourhood of $P$.
Firstly I translated the axes (with the substitution $u=x-1$), and then I tried to use spherical coordinates. I guess $P=(1,0,0)$ is a local minimum, but I'm not sure. I would be grateful if someone could show me how to rigorously prove the nature of this critical point. Thank you so much for the support.

Answer 1

Hint:

Define the curve $\mu(t) = (1,0,|t|^{3/2})$. Now, investigate the composite functions $f \circ \mu$ near $t=0$.