Does a Minimum Exist for x^2+y^2+z^2 Given x^4+y^4+z^4=3?

[arcyqwerty]

Homework Statement

Find max/min of x^2+y^2+z^2 given x^4+y^4+z^4=3

Homework Equations

Use of gradient vectors related by LaGrange Multiplier

The Attempt at a Solution

$$\begin{gathered} f\left( {x,y,z} \right) = {x^2} + {y^2} + {z^2};g\left( {x,y,z} \right) = {x^4} + {y^4} + {z^4} - 3 = 0 \\ \vec \nabla f = \left\langle {2x,2y,2z} \right\rangle ;\vec \nabla g = \left\langle {4{x^3},4{y^3},4{z^3}} \right\rangle \\ \left\langle {2x,2y,2z} \right\rangle = \lambda \left\langle {4{x^3},4{y^3},4{z^3}} \right\rangle \\ 2{x^2} = 2{y^2} = 2{z^2} \to x = \pm y = \pm z \\ 3{x^4} - 3 = 0 \to {x^4} = 1 \to x = \pm 1 \to y = \pm 1,z = \pm 1 \\ \max = f\left( {1,1,1} \right) = f\left( {1,1, - 1} \right) = f\left( {1, - 1,1} \right) = f\left( {1, - 1, - 1} \right) = \\ f\left( { - 1,1,1} \right) = f\left( { - 1,1, - 1} \right) = f\left( { - 1, - 1,1} \right) = f\left( { - 1, - 1, - 1} \right) = 3 \\ \end{gathered}$$

So I found the maximum but does the minimum exist?

 

[Ray Vickson]

Homework Statement

Find max/min of x^2+y^2+z^2 given x^4+y^4+z^4=3

Homework Equations

Use of gradient vectors related by LaGrange Multiplier

The Attempt at a Solution

$$\begin{gathered} f\left( {x,y,z} \right) = {x^2} + {y^2} + {z^2};g\left( {x,y,z} \right) = {x^4} + {y^4} + {z^4} - 3 = 0 \\ \vec \nabla f = \left\langle {2x,2y,2z} \right\rangle ;\vec \nabla g = \left\langle {4{x^3},4{y^3},4{z^3}} \right\rangle \\ \left\langle {2x,2y,2z} \right\rangle = \lambda \left\langle {4{x^3},4{y^3},4{z^3}} \right\rangle \\ 2{x^2} = 2{y^2} = 2{z^2} \to x = \pm y = \pm z \\ 3{x^4} - 3 = 0 \to {x^4} = 1 \to x = \pm 1 \to y = \pm 1,z = \pm 1 \\ \max = f\left( {1,1,1} \right) = f\left( {1,1, - 1} \right) = f\left( {1, - 1,1} \right) = f\left( {1, - 1, - 1} \right) = \\ f\left( { - 1,1,1} \right) = f\left( { - 1,1, - 1} \right) = f\left( { - 1, - 1,1} \right) = f\left( { - 1, - 1, - 1} \right) = 3 \\ \end{gathered}$$

So I found the maximum but does the minimum exist?

Is the feasible set S = {(x,y,z): x^4 + y^4 + z^4 = 3} compact? Is the function f(x,y,z) = x^2 + y^2 + z^2 continuous on S? Have you heard of Weierstrass' Theorem?

RGV

 

[arcyqwerty]

Is the feasible set S = {(x,y,z): x^4 + y^4 + z^4 = 3} compact? Is the function f(x,y,z) = x^2 + y^2 + z^2 continuous on S? Have you heard of Weierstrass' Theorem?

RGV

I'm not quite sure what you mean by compact or Weierstrass' Theorem but I think that the function is continuous

 

[Ray Vickson]

I'm not quite sure what you mean by compact or Weierstrass' Theorem but I think that the function is continuous

Google is your friend.

RGV

 

[arcyqwerty]

Google is your friend.

RGV

So...
"A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S."

Not quite sure what that means exactly, but perhaps its compact if there can be a finite subset of the points defined by the function?

And...
There seems to be two different Theorems, one about estimating functions with polynomials and another about sequence convergence...

 

[Ray Vickson]

So...
"A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S."

Not quite sure what that means exactly, but perhaps its compact if there can be a finite subset of the points defined by the function?

And...
There seems to be two different Theorems, one about estimating functions with polynomials and another about sequence convergence...

If you keep searching you will eventually find a document in which all this is put into the context of ordinary 3-D space with the usual distance measure. In that case there is a theorem saying that a set is compact if and only if it is closed and bounded. So, is the set S closed (i.e., contains all its limit points)? Is it bounded? Then there is a theorem of Weierstrass saying that a continuous function on a compact set assumes both its maximum and its minimum. (These are theorems that are proven in advanced Calculus classes, well before 'topology'.) So, in your case the answer is YES: S is compact, and f has a minimum on S, as well as a maximum. None of this helps you *find* the minimum, but it does tell you that the search makes sense.

RGV