Non-Linear regression

Question

Imagine that I have a function $ f(x,y) $ to model a physical phenomenon.

I believe that functions is defined by $$ f(x,y) = A*x + B*y + C*x*y$$

I have many values for $ (x,y,f(x,y)) $, how can I find the approximated value for $(A,B,C)$?

There are the values:

f(x,y)  |   x   |   y
--------+-------+-------
874     |  75   |  7
954     |  75   |  4
2696    |  75   |  30
2731    |  75   |  29
4227    |  343  |  4
4504    |  75   |  4
4714    |  343  |  7
4962    |  75   |  7
6592    |  75   |  29
6718    |  75   |  30
9419    |  343  |  29
9770    |  343  |  30
12982   |  639  |  4
16184   |  639  |  7
19898   |  1389 |  7
24528   |  1389 |  4
29382   |  639  |  29
29638   |  2101 |  4
30095   |  2274 |  7
30297   |  2101 |  7
30876   |  2274 |  4
31856   |  639  |  30
35121   |  1389 |  30
38708   |  1389 |  29
50192   |  3676 |  4
50482   |  3676 |  7
60837   |  2274 |  30
61340   |  2101 |  29
61850   |  2101 |  30
64533   |  2274 |  29
104421  |  3676 |  29
105073  |  3676 |  30

Answer 1

Do like with linear regression: minimize $$ \sum_i (f(x_i,y_i) - Ax_i - By_i - Cx_iy_i)^2 $$ on $A,B,C$.