In this article at page 924, toward the end of the page, it is shown that the function $F(x)$, $F: G\subset \mathbb{R}^n \to \mathbb{R} $ (the definition of the function is not important for this question) has a minimum at $\bar x$, and then claimed that $F$ has zero partial derivatives at $\bar x$; However, we are working on a compact domain, in particular a closed ball around $x_0 = (0,...,0)$, and we do not have to have a zero partial derivative in the case if the extremum point is not a critical point, which is not something that the author argues that it is the case.
For example, consider $f(x,y) = x^2 + y^2$, and the region $G = \{(x,y)\in \mathbb{R}^2 | 2 \leq x^2 + y^2 \leq 4\}$. The at the minimal and maximal points, this function has non-zero partial derivatives.
So my question is that, firs of all, is there anything wrong with my argument, and secondly, how can the author claims that $F(x)$ has zero partial derivatives at $\bar x$ ?