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Given the function $f(x,y) := x^2y$, which is defined on all $\{(x,y):x^2+y^2\le1\}$, the first partial derivatives are $0$ where $x=0$, however, the Hessian at all these points has the Eigenvalue $0$, it is semidefinite. We are supposed to find all local and global extrema. How does one handle the points where the Hessian is semidefinite? Do they have a name? Is it simply impossible to make a statement on local maxima/minima inside the compact set?

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    $\begingroup$ Another thing to keep in mind: global and local maxima and minima might occur on the boundary of your constraint set, and thus not be found by the usual calculus recipes. $\endgroup$ – kimchi lover Jun 6 at 13:15
  • $\begingroup$ @kimchi lover Thanks for the hint, but should I include or exclude the (0,y) values from my solution? $\endgroup$ – Ruben Kruepper Jun 6 at 13:19
  • $\begingroup$ Your title, question, and comment are inconsistent. Are you asked for local/global extrema, or critical points, or both? Critical points, at least, you can find by calculus. $\endgroup$ – kimchi lover Jun 6 at 13:23
  • $\begingroup$ @kimchi lover Apologies, English is not the language I'm studying in. $\endgroup$ – Ruben Kruepper Jun 6 at 13:25
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We have $f(t,t)=t^3$ and $f(0,t)=0$.

If $t>0$ we get $f(t,t)>f(0,t)$ and if $t<0$ we get $f(t,t)<f(0,t)$.

Conclusion: in $(0,t)$ the function $f$ has no local extremum

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I think you must search the optimum on the vurve $$x^2+y^2=1$$ so you will get $$f(\pm\sqrt{1-y^2},y)=(1-y^2)y$$

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  • $\begingroup$ Yes, but do I include the points (0,y) from the inside of the compact set in my solution set? They may be local extrema, and they may not be... Greetings from Germany. :) $\endgroup$ – Ruben Kruepper Jun 6 at 13:38
  • $\begingroup$ One moment please, i will check my computation. $\endgroup$ – Dr. Sonnhard Graubner Jun 6 at 13:39
  • $\begingroup$ I this case you will get the local extrema $\endgroup$ – Dr. Sonnhard Graubner Jun 6 at 13:42
  • $\begingroup$ The global extrema are situated on the curve $$x^2+y^2=1$$ $\endgroup$ – Dr. Sonnhard Graubner Jun 6 at 13:43

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