What you wrote is different from what the analysis said. The analysis shows that f(x, y) takes the minimum at (x0, y0), f(x, y0) takes the minimum at x=x0, and f(x0, y) takes the minimum at y=y0. This is an affirmative proposition, which is correct.

What you wrote is its inverse proposition: if f(x, y0) takes the minimum at x=x0 and f(x0, y) takes the minimum at y=y0, then f(x, y) takes the minimum at (x0, y0), which is not necessarily true.