Understanding liquid-liquid phase separation

The above figure shows how the regular-solution free energy might lead to liquid–liquid equilibrium. In the region between the compositions $x^I_A$ and $x^{II}_A$, $G$ increases with $x_A$. If the overall concentration lies between these two values, it is always favorable for the
system to spontaneously phase-separate into two regions, one with composition $x^I _A$ and
another with composition $x^{II} _A$. The common tangent line gives the free energy of the phase-separated state.

Okay, so if I have solution with composition between $x^I_A$ and $x^{II}_A$, it will phase separate into whichever potential well they are closes to. That makes sense.

The above graph is just for illustration purposes. It is not directly related to the problem I have.

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My question is, if I have an expression for the excess Gibbs energy of the Margules kind, given by,
$$G^{ex} = RT(c x_1 + kc x_2)x_1x_2$$
how do I find the region where the solution will spontaneously phase separate?
I know $G = G^{ideal}+G^{ex}$, so $G = mu_1 + mu_2 + RTln x_1 + RTln x_2 + G^{ex}$.

I believe that solution will most definitely spontaneously phase out when
$$frac{partial ^2 G}{partial x_1^2} < 0 $$
I can solve for values of $c$ for which the solution separates.

If, however, $c$ is dependent on temperature, such as $c=wT$, how would I find the critical composition and temperature? Would I still impose
$$frac{partial ^2 G}{partial x_1^2} < 0? $$
But that is only one constraint I have, but I need to solve for $T_c$ and $x_c$.

How do I go ahead with this problem?