Local Maxima and Minima of Functions with Two Variables

Example: Locate all critical points and identify local maxima, local minima, and any

saddle points for the function

f(x,y) = (1/3) x³ + xy² ˗ 2xy

Strategy:

Locate critical points, (a,b)	∂f/∂x = x^2 + y^2 ˗ 2y = 0 (equation 1) and ∂f/∂y = 2 xy ˗ 2x = 0 2x(y ˗ 1) = 0 (equation 2)
Use equation 2. Case 1: x = 0 Put into equation 1.	y(y ˗ 2) = 0 Therefore: y = 0 and y = 2 Critical point are: (0,0) and (0,2)
Use equation 2. Case 2: y = 1 Put into equation 1.	x^2 ˗ 1 = 0 So x = ± 1 Critical points are: (1,1) and ( ˗ 1,1)
Calculate D(a,b)	fxx = 2x , fyy = 2x, fxy = 2y ˗ 2 and 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	fxy	fyy	D	Type
0,0	0	˗ 2	0	˗	saddle
0,2	0	2	0	˗	saddle
1,1	2	0	2	+	local min
˗1,1	˗ 2	0	˗2	+	local max

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