At how many of these points is f(x,y) (strictly) greater than both g(x,y) and h(x,y)? and

Question

I try too find out the solution but is not work. Please help!

This what I wrote:

Answer 1

First, define functions.

Next, use Outer to get the function values as matrices..

Matrices can be compared elementwise.

Answer 2

Posting code rather than images helps people providing answers.

There are syntax errors in the image of the code. sin should be Sin.

Here is an approach defining functions (as suggested by @Alan) and using Tuples, though Outer and matrices could be used.

The solution to the first question can be plotted to confirm result. I leave the second question to OP. I hope this is instructive:

f[x_, y_] := x^3 + y^2 - 30 x y - 2
g[x_, y_] := x Sin[x + y] + 6 y
h[x_, y_] := (x^3 + y^3)/(x^2 + Exp[y/100])
mesh = Tuples[Range[-1, 1, 0.1], 2];
fm = f @@@ mesh;
gm = g @@@ mesh;
hm = h @@@ mesh;
fg = Sign[fm - gm] /. -1 -> 0;
fh = Sign[fm - hm] /. -1 -> 0;
pos = Position[fg fh, 1] ;
ans = Extract[mesh, pos]
Length[ans]/Length[mesh]
Show[Plot3D[{f[x, y], g[x, y], h[x, y]}, {x, -1, 1}, {y, -1, 1}, 
  Mesh -> None, PlotRange -> Full, PlotLegends -> "Expressions"], 
 Graphics3D[{Red, Point[{#1, #2, f[#1, #2]} & @@@ ans]}]]

Answer 3

Another option

f[x_, y_] := x^3 + y^2 - 30*x*y - 2;
g[x_, y_] := x*Sin[x + y] + 6*y
h[x_, y_] := (x^3 + y^3)/(x^2 + Exp[y/100]);

pointsf = Table[f[x, y], {x, -1, 1, 0.1}, {y, -1, 1, 0.1}] // Flatten;
pointsg = Table[g[x, y], {x, -1, 1, 0.1}, {y, -1, 1, 0.1}] // Flatten;
pointsh = Table[h[x, y], {x, -1, 1, 0.1}, {y, -1, 1, 0.1}] // Flatten;

pointsf // MapIndexed[(# > pointsg[[First@#2]] && # > pointsh[[First@#2]]) &] // Counts

(* <|False -> 289, True -> 152|> *)