Optimizing Quadratic Functions with the Second Derivative Test

[Zyuke]

Under what conditions on the constants a and b does the second derivative test guarantee that the function

g(x,y,z)=ax^2+2axz+by^2-2byz+z^2

has a local maximum at (0,0,0)? a local minimum at (0,0,0)?

well, i used the Hessian matrix to compute the eigenvalues to set them above zero. but the computation is so complicated that i used Mathematica to solve them and it turned out to be some messy stuff. i was convinced that this is not the way to do it. but then how?

 

[Jufro]

[STRIKE]Just looking at the problem makes me want to try Lagrange multipliers.[/STRIKE]

 

[micromass]

[STRIKE]Just looking at the problem makes me want to try Lagrange multipliers.[/STRIKE]

Won't work here.

They ask about the second derivative test, so you'll need to calculate the Hessian. It's what they ask for.

 

[Ray Vickson]

Under what conditions on the constants a and b does the second derivative test guarantee that the function

g(x,y,z)=ax^2+2axz+by^2-2byz+z^2

has a local maximum at (0,0,0)? a local minimum at (0,0,0)?

well, i used the Hessian matrix to compute the eigenvalues to set them above zero. but the computation is so complicated that i used Mathematica to solve them and it turned out to be some messy stuff. i was convinced that this is not the way to do it. but then how?

Why would you try to set the eigenvalues to zero? The Hessian is positive-definite if all the eigenvalues are > 0 (so the origin is a global minimum---not just local!)

The Hessian cannot be negative-definite, so the origin cannot be a local maximum. (Note: a matrix A is negative-definite if and only if its negative B = -A is positive-definite, and since B(3,3) = -2 < 0, B cannot be positive-definite---no fancy tests needed here!)

To test for positive-definiteness of the Hessian H, just perform Cholesky decomposition; see, eg., http://en.wikipedia.org/wiki/Cholesky_decomposition . The Cholesky algorithm is easy to carry out in this case.

 

[Zyuke]

thanks a lot.