# Max/min with constraints

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?

Homework Helper
Dearly Missed

## 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

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

RGV

arcyqwerty

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...