Monotonicity of implicitly defined function

Question

Let $f(x,y):\mathbb{R}^2\rightarrow\mathbb{R}$ and $g(y):\mathbb{R}^2\rightarrow\mathbb{R}$ be $C^2$-differentiable functions. Let $f(x,y)$ and $g(y)$ be strictly decreasing in $y$, and let $f(x,y)$ be also strictly decreasing in $x$. Now let $x(y)$ be implicitly defined by equation $$f(x,y)+g(y)=0.$$ How can I prove that $x(y)$ is strictly decreasing or increasing in $g(y)$? I already have that $x(y)$ is strictly decreasing in $y$ (via implicit differentiation), but I am stuck with the dependence on $g(y)$.

Answer 1

Assume that $g(y_1) < g(y_2)$. Then $y_1 > y_2$ (because $g$ is decreasing), and $$ f(x(y_1), y_1) + g(y_1) = 0 = f(x(y_2), y_2) + g(y_2) > f(x(y_2), y_1) + g(y_1) $$ since $f$ is decreasing in $y$. Therefore $$ f(x(y_1), y_1) > f(x(y_2), y_1) $$ and that implies $x(y_1) < x(y_2)$ since $f$ is decreasing in $x$. So $$ g(y_1) < g(y_2) \implies x(y_1) < x(y_2) \, , $$ i.e. $x(y)$ is (strictly) increasing in $g(y)$.