use lagrange multipliers to find the extrema of f subject to the stated constraints:

f(xyz) =x^2+y^2+z^2
x-y=1

Question

use lagrange multipliers to find the extrema of f subject to the stated constraints:

f(xyz) =x^2+y^2+z^2
x-y=1

anonymous · Answer

this is what i've gotten so far...
$$fx = 2x - \lambda, x = \lambda/2$$
$$fy = 2y+ \lambda, y = -\lambda/2$$
$$z=2z+ \lambda, z = -\lambda / 2$$
$$f \lambda = -x + y +1, \lambda = 1$$

so then if i plug lambda in, i get x = 1/2, y = -1/2, z = -1/2

if i plug it back into the equation, i get 1/4 + 1/4 + 1/4 .... but .. i think i did something wrong / not sure how to go from  here...

blockcolder · Answer

Tip: Use _ to make a subscript (a_i will produce $a_i$).
Where did the $z=2z+\lambda$ come from?

anonymous · Answer

yeah, i just realized that ... lol f_z = 2z, z = 0, sorry about that!

blockcolder · Answer

The rest is correct, so the maximum is 1/2.

anonymous · Answer

the answer is supposed to be
min f(0, -1, 0) = 1 and 4(2,1,0) = 5
i realize those are the numbers you plug into the f(xyz), but how did they get these numbers? O.o

blockcolder · Answer

But the minimum is supposed to be 1/2 since it's smaller than 1.

anonymous · Answer

thats the answer they gave me in the book :(

blockcolder · Answer

Then there's something wrong with the book. No matter what I did, I always got x=1/2, y=-1/2. Sorry if it's not much help.

anonymous · Answer

no worries, maybe the book is wrong ^.^ thank you!!

blockcolder · Answer

You're welcome. :)