Question

Find the value of x, y, z of [2x+y x-y] =[10 -1] [x-z x-y-z] =[2 8]

Answer 1

By Lagrange Multipliers, ∇f = λ∇g.  

==> <yz, xz, xy> = λ<2x, 4y, 6z>.  

Equating like entries yields  

yz = 2λx  

xz = 4λy  

xy = 6λz.  

Hence, xyz = 2λx^2 = 4λy^2 = 6λz^2.  

If λ = 0, then xyz = f(x,y,z) = 0.  

Otherwise, we have x^2 = 2y^2 = 3z^2 ==> x^2 + x^2 + x^2 = 24 ==> x^2 = 8.  

So, we have 8 critical points: (±√8, ±2, ±√(8/3)), counting all signage possibilities.  

Hence, the extreme values of f are  

f((√8, 2, √(8/3)) = 16/√3 (or when we have an even number of - signs),  

f((-√8, -2, -√(8/3)) = -16/√3 (or when we have an odd number of + signs).  

I hope this helps!