Multivariate Calculus - Bruce E. Shapiro

LECTURE 14. UNCONSTRAINED OPTIMIZATION 115
Now we make the following substitutions,
A = 1 2 g uu,0 = 1 2 f xx,P
B = g uv,0 = f xy,P
C = 1 2 g vv,0 = 1 2 f yy,P
which gives us
g(u, v) = Au 2 + Buv + Cv 2
In other words, we have approximated the original function with a quadratic at the
origin (note to physics majors: this is a generalization of a property called Hooke’s
law to 2 dimensions).
so that
The function
D(u, v) = g uu g vv − (g uv ) 2
d = (2A)(2C) − B 2 = 4AC − B 2
Completing the squares in the formula for g,
g(u, v) = A
[(u 2 + B A uv + B2 v 2 )
4A 2 − B2 v 2
4A 2 + C ]
A v2
[ (
= A u + B ) 2 ( ) ]
C
2A v +
A − B2
4A 2 v 2
[ (
= A u + B ) 2 ( 4AC − B
2
2A v +
= A
[ (
u + B 2A v ) 2
+ d
4A 2 v2 ]
4A 2 )
v 2 ]
We have three cases to consider, depending on the value of d.
Case 1: d > 0
If d > 0 then everything in the brackets is positive except at the origin where it is
zero. Thus the origin is a local minimum of everything inside the square brackets.
The function g is a paraboloid that extends upwards around the origin when A > 0
and downwards when A < 0. Thus if A > 0, we have a local minimum and if A < 0
we have a local maximum.
Math 250, Fall 2006 Revised December 6, 2006.