Principles of Mathematics in Operations Research

72 5 Positive Definiteness

Example 5.1.1 Let f(x,y) = x^2 + y^2. Find the extreme points of f(x,y):

ox ay

Since we have only one critical point, it is either the maximum or the min-

imum. We observe that f(x,y) takes only nonnegative values. Thus, we see

that the origin is the minimum point.

Fig. 5.2. Plot of f(x, y) = xy - x^2 - y^2 - 2x - 2y + 4

Example 5.1.2 Find the extreme points off(x,y) = xy-x^2 —y^2 -2x — 2y+4.
The function is differentiate and has no boundary points.

(^1) dx v dy y
Thus, x = y = — 2 is the critical point.
8^2 f(x,y) nd^2 f(x,y) d^2 f(x,y) 1
Jxx ~ dx (^2) ~ dy (^2) ~Jvy' Jxy ~ Dxdy ~
The discriminant (Jacobian) of f at (a,b) = (—2,-2) is
— fxxfyy — fXy =4—1 = 0.
Since fxx < 0, fxxfyy — f^2 y > 0 =>• / has a local maximum at (—2, —2).
Jxx Jxy
Jxy fyy