Question 1 of 3

(Optimization on triangular region) Find the absolute maximum and minimum values off on the set D when f(x, y) = 3+xy-x-2y, and D is the

closed triangular region with vertices (1, 0), (5,0), and (1,4).

Hint: How do you find out what multivariable function values are on a line segment, i.e., what is the relationship between the variables x and y?

(Optional, but recommended: make a contour plot of f and then look where the triangle is in the xy-plane. Do your answers match what the plot is suggesting?)

finding the derivative :

fx(x,y)=a/ax(3+xy-x-2y)=y-1

fy(x,y)=a/ay(3+xy-x-2y)=x-2

for fx=0

y-1=0-->y=1

for fy=0

x-2-0-->x=2

A(1,0)

B(5,0)

C(1,4)

f(x,y) is (2,1)

f(2,1)=3+(2)(1)-2-2(1)=3

Fxx at (2,1)

Fxy

Fyy*

f(1,0)=3+(1)(0)-1-2(0)=2

f(5,0)=3+(5)(0)-5-2(0)=-2

f(1,4)=3+(1)(4)-1-2(4)=-4

local min = 2

local max=-2