sigmoid[x_,alpha_,beta_,k_]:=k/(1+Exp[-alpha-x*beta])
In[]:=
FullSimplify[​​Reduce[​​{sigmoid[-1,alpha,beta,k]==a,​​sigmoid[0,alpha,beta,k]==b,​​sigmoid[1,alpha,beta,k]==c,​​(0<a<b<c)||(0<c<b<a)​​},​​{alpha,beta,k},Reals]​​]​​
In[]:=
alphaLog
a
a-b
+
b
-b+c
&&betaLog
(a-b)c
a(b-c)
&&
b(a(b-2c)+bc)
2
b
-ac
k&&a>0&&c<
2
b
a
&&((b>0&&c>0&&b<a)||(b>a&&b<c))
Out[]=