mathematical optimization - How to define and plot a maximum function?

Define a function $g$ as follows:

f = 2/((x1 + Sqrt[3] x2)^2 + (y1 + Sqrt[3] y2)^2)
X = vx1^2*D[D[f, x1], x1] + vx2^2*D[D[f, x2], x2] + 2 vx1*vx2*D[D[f, x1], x2] 
Y= vx1*D[f, x1] + vx2*D[f, x2] 
g=X+Y^2

The function $g$ is clearly a function in 6 variables ($vx1,vx2,x1,x2,y1,y2$).

Now I want to define a function in two variables $max$ as the maximum of $g$ for $x1,x2,y1,y2$ which vary in the following hypersurface

H = ImplicitRegion[{3 (-x1^2 - y1^2 + x2^2 + y2^2) + 
 2 Sqrt[3] (x1*x2 + y1*y2) == 0}, {x1, y1, x2, y2}]

So I want to (but I don't know how to) define the function $max:\mathbb{R}^2\rightarrow \mathbb{R}$ as follows:

$$max(vx1,vx2)=\max\limits_{(x1,x2,y1,y2)\in H}g(vx1,vx2,x1,x2,y1,y2)$$

All I could think of is the command

FindMaximum[{g, {x1, y1, x2, y2} ∈ H}, {x1, y1, x2, y2}]

but this doesn't define a function, it just gives the maximum of $g$ for specific values of $vx1,vx2$.

How can I define the function $max$? It's important to me to define it as a function because I need to

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  derive it in its smooth points

  
  


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  plot it

  
  



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  define other functions which involve $max$ in their definition

  
  

How can I do it?

EDIT: I followed the suggestions of user Corey979 and defined the function max[vx1,vx2], but with FindMaximum substitued by MaxValue, since I'm interested in the maximum value (as I thought was clear in my definition). But then I'm not able to do any operation with the function max:

Any of these operations

D[max[vx1, vx2], vx1]

Plot3D[max[vx1, vx2], {vx1, -10, 10}, {vx2, -10, 10}]


FindMaximum[{max[vx1, vx2]}, {vx1, vx2}]

Will require an extremely long computational time and will give no output. Which is strange, since with my old code

f = 2/((x1 + Sqrt[3] x2)^2 + (y1 + Sqrt[3] y2)^2)
X = vx1^2*D[D[f, x1], x1] + vx2^2*D[D[f, x2], x2] + 2 vx1*vx2*D[D[f, x1], x2] 
Y= vx1*D[f, x1] + vx2*D[f, x2] 
g=X+Y^2
H = ImplicitRegion[{3 (-x1^2 - y1^2 + x2^2 + y2^2) + 
 2 Sqrt[3] (x1*x2 + y1*y2) == 0}, {x1, y1, x2, y2}]

FindMaximum[{g, {x1, y1, x2, y2} ∈ H}, {vx1,vx2,x1, y1, x2, y2}]

I was able to get the answer

{2.39111, {vx1 -> 1.55608, vx2 -> 1.96316, x1 -> 0.810245, y1 -> -0.236216, x2 -> 1.34919, y2 -> -0.463331}}

Is there a better way to define max in such a way that it will be possible to perform operations with it?

Answer

Define your functions as functions:

f[x1_, x2_, y1_, y2_] := 
 2/((x1 + Sqrt[3] x2)^2 + (y1 + Sqrt[3] y2)^2)


X[vx1_, vx2_, x1_, x2_, y1_, y2_] := 
 vx1^2*D[D[f[x1, x2, y1, y2], x1], x1] + 
  vx2^2*D[D[f[x1, x2, y1, y2], x2], x2] + 
  2 vx1*vx2*D[D[f[x1, x2, y1, y2], x1], x2]

Y[vx1_, vx2_, x1_, x2_, y1_, y2_] := 
 vx1*D[f[x1, x2, y1, y2], x1] + vx2*D[f[x1, x2, y1, y2], x2]

g[vx1_, vx2_, x1_, x2_, y1_, y2_] := 

 X[vx1, vx2, x1, x2, y1, y2] + Y[vx1, vx2, x1, x2, y1, y2]^2

H = ImplicitRegion[{3 (-x1^2 - y1^2 + x2^2 + y2^2) + 2 Sqrt[3] (x1*x2 + y1*y2) == 0}, {x1, y1, x2, y2}]

Define the max as a function:

max[vx1_, vx2_] := 
 FindMaximum[{g[vx1, vx2, x1, x2, y1, 
    y2], {x1, y1, x2, y2} ∈ H}, {x1, y1, x2, y2}]

Then

m = max[1000., 1000.]

(The value of the maximum is a numerical zero; you might want to incorporate Chop in max.) Verify that the solution is in H:

m[[2, All, 2]] ∈ H

True