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Find minimum and maximum value of function $$f(x,y,z) = x^3 + y^3 + z^3$$ on set $$ \left\{ (x,y,z): x^2 + y^2 + z^2 = 1 \wedge x+y+z = \sqrt{3} \right\} $$

I don't know what is this set. We have sphere and plane so I suppose that it may be circle or point. How find it?

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  • $\begingroup$ Are you familiar with the method of Lagrange multipliers? $\endgroup$ – M. Strochyk Sep 15 '13 at 11:26
  • $\begingroup$ Yes, I'm. I supposed that this method will fail. But I used it, and received that we have only one critical point: $(\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3})$. $\endgroup$ – Thomas Sep 15 '13 at 16:08
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note $$(x^2+y^2+z^2)(1+1+1)\ge (x+y+z)^2$$ so if $$x^2+y^2+z^2=1,x+y+z=\sqrt{3}$$ then we have $$3(x^2+y^2+z^2)=(x+y+z)^2$$ $$\Longrightarrow x^2+y^2+z^2-xy-yz-xz=0$$ so $x=y=z$

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The mean of the square is greater or equal to the square of the mean, with equality only when all the values are equal. According to the constraint, both the mean of the square and the square of the mean are equal to $\frac13$, so the only values consistent with the constraint are x=y=z.

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