Global Maxima and Minima of Functions with Two Variables

Example: Find the absolute maxima and minima for the function, f(x,y) given below

For the domain |x| ≤ 1 and |y| ≤ 1.

f(x,y) = x² + y² + x²y + 9

Strategy:

Locate critical points, (a,b)	∂f/∂x = 2x + 2xy = 0 = 2x(1 + y) (equation 1) and ∂f/∂y = 2 y + x^2 = 0 (equation 2)
Use equation 1. Case 1: x = 0	By equation 2 y = 0 Critical point is: (0,0)
Use equation 1. Case 2: y = ˗ 1	By equation 2 x = ± √ 2 Critical points are: (√ 2, ˗ 1) and ( ˗ √ 2, ˗1) (not in domain)
Calculate D(a,b)	fxx = 2( 1 + y ), fxy = 2 , fxy = 2x D = fxxfyy ˗ fxy^2
Strategy: Set up table to identify local max, local min, and saddle points for f(x,y)

Results

x,y	fxx	fyy	fxy	D	Type
0,0	2	0	2	4 > 0	minimum

Next evaluate values of f(x,y) on the boundaries of domain, D.

Click here to continue.

Return to Calculus 3 Notes

Copyright © 2018 Richard C. Coddington

All rights reserved.