Finding the optimum value of a variable subject to a nonlinear constraint

Question

I want to know whether or not the objective function I have defined is correct. The nonlinear constraint has to be satisfied while ζ varies from 0 < ζ <= 1/2.

This is the first time I am trying to do optimization with Mathematica. It would be helpful if you would direct me to any documentation that can help me.

ClearAll 
f[β_] := β;
(*Objective function*)
a = 1/(2*β^3); 
eq = 
  a*(((Sin[β*(1 - ζ)]*Sin[β*ζ])/Sin[β]) - ((Sinh[β*(1 - ζ)]*Sinh[β*ζ])/Sinh[β])); 
eq1 = (10^12*eq) + 1;
(*This is the constraint equation*)
NMaximize[f[β_], eq1 == 0]

Answer 1

f[β_] = β; (*Objective function*)
a = 1/(2*β^3);
eq = a*(((Sin[β*(1 - ζ)]*Sin[β*ζ])/Sin[β]) - ((Sinh[β*(1 - ζ)]*Sinh[β*ζ])/Sinh[β]));
eq1 = (10^12*eq) + 1; (*This is the constraint equation*)

Use the proper syntax as shown in the documentation for NMaximize: "NMaximize[{f, cons}, {x, y,…}] maximizes f numerically subject to the constraints cons."

NMaximize[{f[β], eq1 == 0, 0 < ζ <= 1/2}, {β, ζ}, 
  WorkingPrecision -> 50] // N

(* {3.92685, {β -> 3.92685, ζ -> 0.00015584}} *)

EDIT: "For nonlinear functions, NMaximize may sometimes find only a local maximum."

NMaximize[{f[β], eq1 == 0, 
   0 < ζ <= 1/2}, {{β, 3, 10}, ζ}, WorkingPrecision -> 50] //
  N

(* {7.07025, {β -> 7.07025, ζ -> 0.000364042}} *)