$Assumptions=c>0&&c<1&&ρ>0&&ρ<1

(*dynamical Landau free-energy for a KCM on a complete graph s conjugated to activity*)
F[ρ_,s_,c_]=c ρ (1-ρ) +(1-c) ρ^2-2 Exp[-s]Sqrt[c (1-c) ρ (1-ρ)]ρ

(*psi_K(s) is given by the minimum of F[ρ,s,c] for ρ on [0,1]*)
Manipulate[
cc=.6;
Plot[F[ρ,s,cc],{ρ,0,1},PlotStyle->{Thick,Darker@Purple},Frame->True,
PlotRange->{{0,1},{-0.04,.12}},FrameLabel->{"ρ","ℱ(ρ,s)"},LabelStyle->Directive[FontSize->20],PlotLabel->"s="<>ToString[N[s,3]]],
{{s,0},-.2,.2}]

ρofs[s_,c_]=Simplify[ρ/.Solve[D[F[ρ,s,c],ρ]==0,ρ][[3]]];

Plot[Piecewise[{{ρofs[s,cc],s≤0},{0,s≥0}}],{s,-1,1},AxesOrigin->{0,0},PlotStyle->{Thick,Darker@Purple},Frame->True,Exclusions->False,
PlotRange->{All,{-0.1,.8}},FrameLabel->{"s","ρ(s)"},LabelStyle->Directive[FontSize->20]]

ψsol[s_,c_]=-F[ρofs[s,c],s,c];

Plot[Piecewise[{{ψsol[s,cc],s≤0},{0,s≥0}}],{s,-1,1},AxesOrigin->{0,0},PlotStyle->{Thick,Darker@Purple},Frame->True,Exclusions->False,
PlotRange->{All,{-0.1,.8}},FrameLabel->{"s","ψ(s)"},LabelStyle->Directive[FontSize->20]]

ψ1sol[s_,c_]=D[ψsol[s,c],s];

Plot[Piecewise[{{-ψ1sol[s,cc],s≤0},{0,s≥0}}],{s,-1,1},AxesOrigin->{0,0},PlotStyle->{Thick,Darker@Purple},Frame->True,Exclusions->False,
PlotRange->{All,{-0.1,.8}},FrameLabel->{"s","-ψ'(s)=(Subscript[⟨K⟩, s])/t"},LabelStyle->Directive[FontSize->20]]

(*dynamical Landau free-energy for a KCM on a complete graph s conjugated to density*)
Fn[ρ_,s_,c_]=c ρ (1-ρ) +(1-c) ρ^2-2 Sqrt[c (1-c) ρ (1-ρ)]ρ +s ρ

(*psi_K(s) is given by the minimum of F[ρ,s,c] for ρ on [0,1]*)
Manipulate[
cc=.6;
Plot[Fn[ρ,s,cc],{ρ,0,1},PlotStyle->{Thick,Darker@Purple},Frame->True,
PlotRange->{{0,1},{-0.04,.12}},FrameLabel->{"ρ","ℱ(ρ,s)"},LabelStyle->Directive[FontSize->20],PlotLabel->"s="<>ToString[N[s,3]]],
{{s,0},-.2,.2}]

ρnofs[s_,c_]=Simplify[ρ/.Solve[FullSimplify[D[Fn[ρ,s,c],ρ]]==0,ρ][[3]]];

Plot[Piecewise[{{ρnofs[s,cc],s≤0},{0,s≥0}}],{s,-.3,.3},AxesOrigin->{0,0},PlotStyle->{Thick,Darker@Purple},Frame->True,Exclusions->False,
PlotRange->{All,{-0.1,.8}},FrameLabel->{"s","ρ(s)"},LabelStyle->Directive[FontSize->20]]

ψnsol[s_,c_]=-Fn[ρnofs[s,c],s,c];

Plot[Piecewise[{{ψnsol[s,cc],s≤0},{0,s≥0}}],{s,-.3,.3},AxesOrigin->{0,0},PlotStyle->{Thick,Darker@Purple},Frame->True,Exclusions->False,
PlotRange->{All,{-0.05,.25}},FrameLabel->{"s","ψ(s)"},LabelStyle->Directive[FontSize->20]]

ψn1sol[s_,c_]=D[ψnsol[s,c],s];

Plot[Piecewise[{{-ψn1sol[s,cc],s≤0},{0,s≥0}}],{s,-.3,.3},AxesOrigin->{0,0},PlotStyle->{Thick,Darker@Purple},Frame->True,Exclusions->False,
PlotRange->{All,{-0.1,.8}},FrameLabel->{"s","-ψ'(s)=(Subscript[⟨ρ⟩, s])/t"},LabelStyle->Directive[FontSize->20]]