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Suppose you have two functions $F$ and $G$ with the following properties.

$G(0)=F(0)=0, G'>0, G''<0, F'>0, F''<0 $ and also $\lim_{x\to0} F'(x)=\infty, \lim_{x\to\infty} F'(x)=0, \lim_{x\to0} G'(x)=0, \lim_{x\to\infty} G'(x)=\infty$

Suppose that $s$ is defined such that:

$\frac{F'(S)}{G'(S)}=ab$ (1)

where $a>0$ and $b\in[0,1]$ (Note, I'm not sure these conditions on a,b matter so much)

Obviously, the above equation yields a solution for $S$ such that $S=S(a,b)$

I have to find $\lim_{a\to\infty} \frac{\partial S}{\partial b}$ and $\lim_{a\to 0} \frac{\partial S}{\partial b}$

My answer so far:

From (1), using implicit differentiation, I get:

$\frac{F''(S)G'(S)-F''(S)G'(S)}{(C'(S))^2}\frac{\partial S}{\partial b}=a$ $\implies \frac{\partial S}{\partial b}= \frac{a(G'(S))^2}{F''(S)G'(S)-G''(S)F'(S)}$ (2)

Also from (1), as $a\to 0, S\to \infty $

However, there are no assumptions made on $F''(x)G'(x)$ so I'm not sure how to continue since this could go to 0 or infinity. Do I need to split the problem into cases?

Edit: Ok I have extra assumptions: Let $F''(x)G'(x)\to 0$ and $G''(x)F'(x)\to 0$ as $x \to \infty$

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