How to solve two implicit functions with two variables and parameters

Question

I have two functions given as;

u[c_,d_] := Sinh[a b^2 c^2 + 3a b d] and v[c_,d_] := Log[a b^2 c^2 + 3a b d + d^2] - d. I want to find the root of these two functions that maximise another function say w[c_,d_]:= 2a b c + 3b c - 2c d^2 + c*Exp[a+b]. Here, a and b are parameters that lie between [-3,3] and [1,2] respectively, while c and d are the variables of interest. I am able to solve for c and d when I use the FindRoot command and fix a and b (example, a = 1 and b = 0.7).

a=1; b=0.7;
FindRoot[{u[c,d]== 0,v[c,d]== 0},{c,0},{d,0}]

When I do this I get only one root of the problem and as I change the initial point I get one of the other roots and so on. And I am also able to determine the c and d that gives the max of w[c,d]. But this is what I want to do;

I want to be able to estimate c and d over the entire space of a and b and find the point ({c,d}) which maximises w[c,d]. Also, to plot the surface of u[c,d] with the c and d that gives the maximum of w[c,d].

I am new to mathematical and your help is much appreciated.

Answer 1

u[c_, d_] := Sinh[a b^2 c^2 + 3 a b d];
v[c_, d_] := Log[a b^2 c^2 + 3 a b d + d^2] - d;
w[c_, d_] := 2 a b c + 3 b c - 2 c d^2 + c*Exp[a + b];
NMaximize[{w[c, d], u[c, d] == 0, v[c, d] == 0, -3 <= a <= 3, 
  1 <= b <= 2}, {a, b, c, d}]

{98.4569, {a -> 2.99992, b -> 1.14721, c -> 1.35641, d -> -0.703619}}