A maximum or minimum within the explored domain

If there is a true optimum within the explored domain there will be a turning point on the response surface which will either be a maximum or a minimum. A tangential plane to the surface at this point will have a zero slope in all directions. This is equivalent to saying that the partial derivatives of the response surface model with regard to all experimental variables will be zero at this point. 

The above relation defines a system of equations and its solution gives the coordinates of the maximum (or minimum) point, see Fig. 12.6a. A point on the response surface in which all partial first derivatives are zero will hereafter be called a stationary point. 

A stationary point does not always correspond to the optimum conditions. It can be a saddle-point (minimax) at which the surface passes through a maximum in certain directions, and through a minimum in other directions. Saddle points are rather conunon. To improve (increase) the response, the directions in which the response surface increases should be explored. 

The nature of a stationary point is conveniently determined by the canonical analysis. 

As an example of how the optimum conditions are determined from the response surface model, the enamine synthesis described above is used. It was seen in Fig 12.5 that there was a maximum in the explored domain. The coordinates for the maximum point are computed as follows  

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Fig.12.6 Stationary points on response surfaces (a) A maximum or minimum within the explored domain (b) A maximum or minium outside the explored domain.

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