Finding two dimensional critical point

I'm reading an article about bi layered membranes which state that for the free energy function
$f(\theta) = \theta \ln \theta + (1-\theta)\ln(1-\theta) + \chi \theta (1-\theta)$

Where $\phi_i$ is the mole fraction of certain types of lipids in layer $i$ and $\chi$ is the energy of interaction between two lipids molecules composing the layer. The article states that a critical point exist at $\chi=2$ and $\theta=0.5$. Reading in wikipedia, I concluded that removing the layers interaction part, I simply need to take the second and third derivative of the free energy of the membrane. How do I do this with the interactions? Where can I read about this?

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Which article are you reading? –  Qmechanic Oct 6 '13 at 9:11

What do you mean by “removing the interactions”? The critical point is the point where the second and third derivatives of $f(\theta)$ are equal to zero. This gives directly the conditions you cited: