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I'm looking for the minimum of this function for real numbers, I proved that the only possible local minimum is $0$ but I didn't find a way to prove/disprove that it's a global minimum.

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    $\begingroup$ Are the variables assumed to positive? $\endgroup$ – Dr. Sonnhard Graubner Jun 26 at 14:38
  • $\begingroup$ no the function is defined for all reals $\endgroup$ – wostysums Jun 26 at 14:39
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There is no minimum if $x=y=z=t\rightarrow +\infty$ than your expression is $3t^2-2t^3 \rightarrow -\infty$

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  • $\begingroup$ How does this discount the existence of a local minimum? $\endgroup$ – B. Goddard Jun 26 at 14:46
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    $\begingroup$ OP is looking for a global minimum as far as I can understand from the question. $\endgroup$ – AO1992 Jun 26 at 14:50
  • $\begingroup$ @AO1992 can you tell me how you came up with the idea for the counterexample (or if you know any relevant books that help develop such strategies) ? Or is it actually a standard trick in multivariable Calculus? $\endgroup$ – wostysums Jun 27 at 19:32
  • $\begingroup$ @wostysums it is pretty common to study the behaviour of a multivariable function by checking it on some specific curves. Here you also have a non-homogeneous polynomial which has only one term of degree 3, so this also helps $\endgroup$ – AO1992 Jul 1 at 8:09

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