Multivariate Calculus - Bruce E. Shapiro

116 LECTURE 14. UNCONSTRAINED OPTIMIZATION
Case 2: d < 0
If d < 0 then write d = − |d| = m 2 > 0 for some number m > 0, so that
[ (
g(u, v) = A u + B ) ]
2 ( mv
) 2
2A v −
2A
If we make yet another change of variables,
then
p = u + Bv
2A , q = mv
2A
g(u, v) = A [ p 2 − q 2]
This is a hyperbolic paraboloid with a saddle at (p, q) = (0, 0). The original critical
point is at u = 0, v = 0, which corresponds to (p, q) = 0. So this is also a saddle
point in uv-space.
Case 3: d=0
Finally, if d = 0 then
(
g(u, v) = A u + Bv ) 2
2A
Along the line u = −B/(2A)v, g(u, v) identically equal to zero. Thus g(u, v) is a
constant along this line. Otherwise, g is always positive when A > 0 and always
negative (when A < 0). Thus the origin is neither a maximum of a minimum.
Revised December 6, 2006. Math 250, Fall 2006