<s>[INST] Prove that the minimum values of $x^2+y^2+z^2$ is $27$ with given condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$. Question: Prove that the minimum values of $x^2+y^2+z^2$ is $27$, where $x,y,z$ are positive real variables satisfying the condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$.  
From AM$\ge$ GM, we have  $\left( \dfrac{x^2+y^2+z^2}{3}\right)^3\ge (xyz)^2=(xy+yz+zx)^2$. Is it possible to show thae result from this relation?
 [/INST] Also, the Tangent Line method helps: $$x^2+y^2+z^2-27=\sum_{cyc}(x^2-9)=$$
$$=\sum_{cyc}\left(x^2-9-18\left(\frac{1}{x}-\frac{1}{3}\right)\right)=\sum_{cyc}\frac{(x-3)^2(x+6)}{x}\geq0.$$
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