# Calculating extreme values for a multivariable function

• Apr 16th 2009, 09:15 PM
Jukodan
Calculating extreme values for a multivariable function
I'm supposed to find the absolute extreme values taken on by f on the set of D.

f (x, y) = 2x^2+y^2−4x−2y+2, D = {(x, y) : 0 ≤ x ≤ 2,0 ≤ y ≤ 2x}.

Could someone please teach me how to do this. The book doesn't really do it for me. I need to figure out how to find the extreme values.
• Apr 16th 2009, 11:01 PM
fardeen_gen
f (x, y) = 2x^2 + y^2 − 4x − 2y + 2
df / dx = 4x -4
df / dy = 2y -2

A) set df /dx = 0
4x -4 = 0
4x = 4
x = 1

B) set df / dy = 0
2y -2 = 0
2y = 2
y = 1
• Apr 16th 2009, 11:54 PM
matheagle
You also need to 'walk' yourself around the boundary, which reduces this to a calculus one problem.
BUT you should always reduce these...

$2(x^2-2x)+y^2-2y+2 = 2(x^2-2x+1)-2+(y^2-2y+1)-1+2 = 2(x-1)^2+(y-1)^2-1$.

That clearly shows that the GLOBAL min is -1 when x=y=1.
And your max will occur somewhere on the boundary.