more tractable for analysis. Factor out the leading term and then complete the
square on the first two terms to obtain
d^2 z
ds^2 =Acos(^2) α+ 2 Bcos αsin α+Csin (^2) α
=A
[
cos^2 α+ 2 B
A
cos αsin α+C
A
sin^2 α
]
=A
[(
cos α+
B
Asin α
) 2
- (AC −B^2 )
A^2 sin
(^2) α
]
.
(7 .74)
Figure 7-15. Saddle point for z(x 0 , y 0 )
One can now make the following observations:
1. If AC −B^2 =zxxzyy −(zxy)^2 = 0,then in those directions αwhich satisfy cos α+
BAsin α= 0 , the second derivative vanishes. For all other values of α, the second
derivative is of constant sign, which is the same sign as A. If the above conditions
are satisfied, then the second derivative test for a maximum or minimum fails.
2. If AC −B^2 =zxx zyy −(zxy)^2 < 0 ,then the second derivative is not of constant sign,
but assumes different signs in different directions α. In particular, for the special
case α= 0 one finds d
(^2) z
ds^2 =Aand for αsatisfying cos α+
BAsin α= 0 there results
d^2 z
ds^2=A(AC −B(^2) )
A^2
sin^2 α.