constrained optimization 3 variables

[burt]

The following image includes the question and my work. The reason why I am posting it here is because I am starting to feel comfortable with 2 variables, but the third one is throwing me off. I am pretty certain about my equations, I just am not sure that my method for solving for my points was valid.
Any input is appreciated!
Thank you for all your help as I am trying to learn this complex subject!

 

[Romsek]

I'm showing a minimum of -12 at

[MATH]\left\{-\sqrt{2},-1,\sqrt{2}\right\},\left\{ \sqrt{2},-1, -\sqrt{2}\right\}[/MATH]
I'm not sure how you've been taught but I've always done Lagrange multipliers as solving

[MATH]\nabla (f+\lambda g) = 0[/MATH]
i.e. you have to combine the function to be maximized and the constraint when taking the gradient and solving.

You seem to do this separately for the two of them.

 

[burt]

∇(f+λg)=0∇(f+λg)=0\displaystyle \nabla (f+\lambda g) = 0

So you add the two functions first and then take the gradient?

 

[burt]

I'm showing a minimum of -12 at

[MATH]\left\{-\sqrt{2},-1,\sqrt{2}\right\},\left\{ \sqrt{2},-1, -\sqrt{2}\right\}[/MATH]

I didn't use these because I found that \(z=x\).
Was that assumption incorrect on my part?

 

[Romsek]

I didn't use these because I found that \(z=x\).
Was that assumption incorrect on my part?

flip the sign on \(\displaystyle \lambda\) and \(\displaystyle z=-x\) becomes a valid solution as well.

 

[Romsek]

So you add the two functions first and then take the gradient?

you want to maximize/minimize \(\displaystyle f(x,y,z)\) subject to \(\displaystyle g(x,y,z) = 0\)

Solve \(\displaystyle \nabla (f + \lambda g) = 0\)

and choose the critical points that produce the maximum/minimum

If there is no solution then you need to explore the boundaries as you were doing.

 

[burt]

flip the sign on \(\displaystyle \lambda\) and \(\displaystyle z=-x\) becomes a valid solution as well.

I didn't know that was legal

 

[burt]

you want to maximize/minimize \(\displaystyle f(x,y,z)\) subject to \(\displaystyle g(x,y,z) = 0\)

Solve \(\displaystyle \nabla (f + \lambda g) = 0\)

and choose the critical points that produce the maximum/minimum

If there is no solution then you need to explore the boundaries as you were doing.

I think I understand. Basically, what I was doing was fine - but it's unnecessary work if it can be found in your simpler way.