564 Real Analysis
that the second derivative isstrictlynegative; the case in which it is zero makes the points
collinear, in which case we are done.
Remark.This is the two-dimensional least absolute deviations problem. This method
for finding the line that best fits a set of data was used well before Gauss’ least squares
method, for example by Laplace; its downside is that it can have multiple solutions (for
example, if four points form a rectangle, both diagonals give a best approximation).
The property proved above also holds inndimensions, in which case a hyperplane that
minimizes the sum of distances from the points passes throughnof the given points.
508.We assume that the light ray travels fromAtoBcrossing between media at pointP.
LetCandDbe the projections ofAandBonto the separating surface. The configuration
is represented schematically in Figure 70.
A
B
D
C x y
P
Figure 70LetAP=x,BP=y, variables subject to the constraintg(x, y)=x+y=CD.
The principle that light travels on the fastest path translates to the fact thatxandy
minimize the function
f (x, y)=√
x^2 +AC^2
v 1+
√
y^2 +BD^2
v 2.
The method of Lagrange multipliers gives rise to the system
x
v 1√
x^2 +AC^2=λ,
y
v 2√
y^2 +BD^2=λ,x+y=CD.From the first two equations, we obtain
x
v 1√
x^2 +AC^2=
y
v 2√
y^2 +BD^2