(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For what real values of the parameters a,b,c,d does the functiob [tex] f(x,y)=ax^3+by^3+cx^4+dy^4-(x+y)^5 [/tex] have a local minimum at (0,0)

2. Relevant equations

I calculated the gradient at (0,0) and it is always zero regardless of parameters.

The problem is that the Hessian matrix is also zero so I don't know what kind of criticial point it is.

I also noticed that if (0,0) is a minimum then [tex] ax^3+by^3+cx^4+dy^4>(x+y)^5 [/tex] in the nieberhood but that still hasnt taken me very far.

I don't see how I can use Lagrange multipliers, the inverse or implicit function theorems, since the gradient is 0 which precludes using them in any direct way. So my arsenal seems rather depleted.

Any ideas?

Thanks

Tal

3. The attempt at a solution

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# Homework Help: Finding multivariate extrema with degenerate hessian matrix

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