Question

find the minimum value of x^2+y^2+z^2 with subject to condition xyz=a^3

Answer 1

Answer:

Using the method of Lagrange Multipliers I get the equations as:

1)2x=λ(1)+μ(yz)

2)2y=λ(1)+μ(xz)

3)2z=λ(1)+μ(xy)

Multiplying (1) by x, (2) by y and , (3) by z I get:

2(x2+y2+z2)=λ(1)+μ(3xyz)

[Using 3xyz=−3]

2u=λ−3μ where u is the function to be maximized/minimized. λ and μ are constants.

After this I'm not sure how to proceed to find maximum/minimum value of u.

Step-by-step explanation:

2x−μyz=λ=2y−μxz2(x−y)=μz(y−x)μz=−2, or x=y

And we can do similar algebra to show that

μx(y−z)=2(z−y) and μy(z−x)=2(x−z)

Either x=y or y=z or x=z

If we assume that all are different we would come to the conclusion that μx=μy=μz=−2 creating a contradiction.

if x=y then z=−1x2

2x−1x2=12x3−x2−1=0

which only has one real solution:

(1,1,−1),(1,−1,1),(−1,1,1) is our solution set.

and x2+y2+z2=3

There is no maximum.