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.