Problem $1)$ Incipient miscibility in polymer$-$solvent systems

From the general form of Flory$-$Huggins $(FH)$ equation, we can express the activity of the solvent $(a_1)$ in a solvent$(1)-$polymer$(2)$ mixture by

$ln{a}_1=ln(1-{}_2)+(1-r^{-1}){}_2+{}_2^2$

where ${}_2$ is the volume fraction of the polymer, $$ is the energy$/$interaction parameter $($depends here only on $T),$ and $r$ is the polymer segment number. The requirement for incipient miscibility

$($i$.$e$.$ the critical point of the solution$)$ is that

$\frac{delln{a}_1}{\text{d}\text{e}\text{l}{}_1}=\frac{de{l}^{2}ln{a}_1}{\text{d}\text{e}\text{l}{}_1^2}=0$

$1$a$)$ Demonstrate from $(1-1)$ and $(1-2)$ that the volume fraction of the of polymer at the critical

point is given by

${}_2^c=\frac{1}{1+r^{\frac{1}{2}}}$