Consider the curve defined by $x^2 + 2y^2 + 4 \beta x y = K $ with $K > 0$ and where $\beta$ is a (sufficiently small) parameter. Assuming that the above can be used to define a function $x = G(y)$, use implicit differentiation to find out what is the largest value that $x$ can take on along this curve.
My attempt: Since $x= G(y)$, the largest value of x is obtained when $G(y)$ is maximized, or when $G'(y)= 0$. However, the given equation of the curve isn't $G(y)$ itself. How do I express the curve equation as $G(y)$, or is there another way of finding the largest value of x along this curve?
Hint
If you differentiate the given expression above, say $F(x,y) - K = 0$, with respect to $y$, considering $x = x(y)$, you would have:
$$ 2 \, x(y) \, x'(y) + 4 y + 4 \beta \, (x'(y) \, y + x(y) ) = 0, $$
where use has been made of the chain rule.
If you need more help, let me know.
Cheers!