6.1 In a study to determine the nature of a response system that relates dry modulus
of rupture (psi) in a certain ceramic material with three important independent
variables, the following quadratic regression equation was determined [see
Hackney and Jones (1969)]:
$\hat{y} \times 10^{-2} = 6.88 - 0.1466x_1^2 + 0.1875x_1x_2 + 0.2050x_1x_3 + 0.0325x_1$
$- 0.0053x_2^2 - 0.1450x_2x_3 + 0.2588x_2 + 0.1359x_3^2 - 0.1363x_3$
The independent variables represent ratios of concentration of various ingredients in
the material.
(a) Determine the stationary point.
(b) Put the response surface into canonical form, and determine the nature of the
stationary point.
(c) Find the appropriate expressions relating the canonical variables to the indepen-
dent variables $x_1$, $x_2$, and $x_3$.
(d) Generate two-dimensional graphs showing contours of constant estimated
modulus of rupture. Use $x_3 = -1, 0, 1.$
