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I try to answer this question The Maximum and Minimum Functions of Two Functions
I wrote the following code
f[x_, y_] := 1 + 2*x + 3*y^3
g[x_, y_] := y + x^2
maxi[x_, y_] :=
Refine[{(f[x, y] + g[x, y])/2 + Abs[f[x, y] - g[x, y]]/2},
Assumptions -> {0 <= x <= 1, 0 <= y <= 1}]
Plot3D[maxi[x, y], {x, 0, 1}, {y, 0, 1}]
Is there any way to find function of maxi?
Try
Simplify[(f[x, y] + g[x, y])/2 + Abs[f[x, y] - g[x, y]]/2,0<=x<=1&&0<=y<=1]
which instantly returns
1+2 x+3 y^3
You can see that by inspection because f[x,y]>g[x,y] over the domain so the Abs does nothing and disappears and that leaves f[x,y]/2+g[x,y]/2+f/x,y]/2-g[x,y]/2==f[x,y]
f[x_] := x^2 and g[x_] := 1 - x, maxi1[x_] = Refine[{(f[x] + g[x])/2 + Abs[f[x] - g[x]]/2}, Assumptions -> {0 <= x <= 1}]; and Simplify[maxi1[x], 0 <= x <= 1].
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f[x_] := x^2; g[x_] := 1 - x; list={{f[x],Reduce[f[x]>=g[x]&&0<=x<=1,x]}, {g[x],Reduce[f[x]<=g[x]&&0<=x<=1,x]}}; Piecewise[list] which gives you a Piecewise function that is the maximum of f and g over the domain. The minimum can be obtained in the same way. But I suspect this answer will not be satisfactory for you either.
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