Question

use Lagrange to find Max, min of f(x,y,z) =x^2+y^2+z^2 subject to x+y+z=12. I need help for the last part of considering whether the critical point is max or min. Please, help

Answer 1

I got L =8 and critical point is (4,4,4) just one point. and don't know how to consider whether it's max or min

Answer 2

use the second partial derivatives of x and y and z. If these are positive you have min...and If these are negative you have max.

Answer 3

I know how to get it with 2 variables only by using formula D = fxx*fyy-(fxy)^2. this is 3variables function.

Answer 4

bad thing about Lagrange multiplies ... they don't tell you max or min, just plugin into your critical points into function and find out which is which and which is not which.

Answer 5

get the hessian determinant

Answer 6

@experimentX hihi, it is. but the question ask me find it out. no comment.

Answer 7

http://androulakis.bma.upatras.gr/mediawiki/images/math/9/9/b/99bf6d18092eab3f8e35007675e6c64b.png use it ;)

Answer 8

@Kate.Ch. take determinant of three element whose entries is fxx, fyy, fzz?

Answer 9

but how to arrange the non diagonal entries?

Answer 10

@Kate.Ch. I got it. let me try. unfortunately, mine is even problem in book, no way to check whether my answer is right or wrong. it's extra assignment I give myself. Thanks anyway

Answer 11

\[\left[\begin{matrix}f(xx) & f(xy) & f(xz)\\f(yx) & f(yy) & f(yz)\\ f(zx) & f(zy) & f(zz)\end{matrix}\right]\]is it right? long long long way

Answer 12

and after i got the det of that matrix, if it is >0, consider fxx>0 or fxx<0 to know it is max or min? how about fyy, fzz? ignore them?