# Minimum of $x^2+y^2+z^2-2xyz$

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.

• Are the variables assumed to positive? – Dr. Sonnhard Graubner Jun 26 at 14:38
• no the function is defined for all reals – wostysums Jun 26 at 14:39

## 1 Answer

There is no minimum if $$x=y=z=t\rightarrow +\infty$$ than your expression is $$3t^2-2t^3 \rightarrow -\infty$$

• How does this discount the existence of a local minimum? – B. Goddard Jun 26 at 14:46
• OP is looking for a global minimum as far as I can understand from the question. – AO1992 Jun 26 at 14:50
• @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? – wostysums Jun 27 at 19:32
• @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 – AO1992 Jul 1 at 8:09