Find Critical Points of a Quadratic Form

Question

Define $f: \mathbb{R}^n \to \mathbb{R} $ for $ n > 1$, $$ f(x) = \frac{1}{2} z^T Q z $$

where $ x = [x_1, \dots, x_n]^T \in \mathbb{R}^n $, $ z = [x^T, y]^T \in \mathbb{R}^{n+1} $ and $$ y = \dfrac{1}{\prod_{i = 1}^{n} x_i} \in \mathbb{R} $$
and $$ Q = 1 1^T - I \in \mathbb{R}^{(n+1) \times (n+1)} $$

with $1$ as a vector of ones and $I$ the identity matrix.

1. Find the critical point $x^*$ of $f(x)$ for witch $x_i > 0$ for all $i$.
2. Verify that $x^*$ is minimum.
3. Compute $f(x^*)$

My first athem was to calculate the gradient as a usual quadratic form $ \nabla f(x) = Qz $ but the vector in the function is Z not x. So I did the multiplication $Z^T Q Z$ to calculate the partial derivate for each $x_i$ but I didn't get an useful expresion.