Anyone want to verify answers for lagrange multiplier question?

Would all be helping me out here alot :)

Find the local maxima and minima of the following function:

$\displaystyle f(x_1,x_2,x_3)=x_1 + 2x_2 + x_3$

Subject to the constraints:

$\displaystyle x_1^2+x_2^2+x_3^2=2$ and, $\displaystyle x_1^2+(x_2-1)^2+x_3^2=3$

The answers I got were:

$\displaystyle (x_1,x_2,x_3)=(1,0,1)$ with $\displaystyle \lambda_1=-\frac{3}{2}, \lambda_2=1$ ... This gives a local maximum.

Also...

$\displaystyle (x_1,x_2,x_3)=(-1,0,-1)$ with $\displaystyle \lambda_1=-\frac{1}{2}, \lambda_2=1$ ... This gives a local minimum.

I hope these are right. If I have made an error, and you want my working just reply... thanks!!

Re: Anyone want to verify answers for lagrange multiplier question?

Did you notice that your second constraint can be nicely simplified with your first constraint? Rather simplifies the whole problem, in fact.

Re: Anyone want to verify answers for lagrange multiplier question?

Yes, I did notice this. And this was my first step used in the problem - that is$\displaystyle x_2=0$

I was only asking to see if anyone can verify my answers....

Re: Anyone want to verify answers for lagrange multiplier question?

Why do you doubt? Did you make any mistakes on purpose?

I'm just questioning why you can't verify your own answers. How did you determine the two points were local max and min?

Re: Anyone want to verify answers for lagrange multiplier question?

Haha, I doubt myself all the time - although the fact that they are nice answers gives me some confidence. I determined local max/min by doing the whole Bordered hessian matrix technique, confirming they weren't non-degenerate critical points, and then checking that $\displaystyle \textbf{h}^TH\textbf{h}$, where $\displaystyle \textbf{h}$ is the tangent vector at that point, was always positive to give a local min, negative to give a local max....

??

Re: Anyone want to verify answers for lagrange multiplier question?

Well, there you go. If you can even SAY "Bordered Hessian" I have to think that you ahve at least a little clue. As long as you used the right criteria for various minors, you have it! Did you? :-)