Showing that the following reasoning is wrong

Question

Suppose we have a function of two variables $f(x,y)$. We know that the function is jointly convex in the feasible set. Furthermore, the feasible set is convex. So the problem of minimizing $f(x,y)$ over the feasible set is a convex optimization problem. Assuming that the constraints are non-active. In this case if we take the derivative of $f(x,y)$ with respect to $x$ and put it equal to zero to get an equation of the form $y=g(x)$. After that we replace $y$ by $g(x)$ in the original objective function $f(x,y)$ (i.e. we write $f(x,g(x))$) and then minimize $f(x,g(x))$ with respect to $x$. What is the way to show that this reasoning is wrong? Thank you.