When marginal profit is equal to zero, the profit curve U
is at a maximum. Therefore, in order to find the amount
of the input which results in maximum profit, the
following relationship is calculated:

 [(dY/dX) Py] Px = 0 or
 dY/dX = Px/Py

In other words, to maximize profit, the derivative of the
response surface (dY/dX) is equated to a ratio of the
price of the input to the price of the product (Px/Py).
The prices of the inputs and the prices of the products
must be measured in the same units used in calculating
regression.
 For two or more inputs, the solution is an
extension of the above:

 U = Y(Py) X1(Pxl) X2(Px2) -...- Xn(Pxn) FC

where X1, X2,..., X, are the inputs included as variables
in the response surface. The partial derivatives (these
are partial derivatives because there is now more than one
input, X) are:

 6U/6X1 = [(6Y/6X1) Py] Pxl = 0
 6U/6X2 = [(6Y/6X2) Py] Px2 = 0




 6U/6Xn = [(6Y/6Xn) Py] Pxn = O

The simultaneous solution of this set of partial
derivatives will result in the combination of the n inputs
to produce product Y that maximizes profit for farmers.

 There are various ways to estimate response surfaces.
The best in any case will depend on the quantity of
inputs, the number of treatments, and the type of
calculating or computing equipment available to
researchers.