<s>[INST] Minimum value of $\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}$ Let $x,y,z>0$ and $x+y+z=xyz$. What is the minimum value of $$\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}?$$
In the case when $x=y=z$, the equation $x+y+z=xyz$ translates to $3x=x^3$, or $x=\sqrt{3}$. If $A$ denotes the quantity that we want to minimize, then $A=\sqrt{3}$ as well. 
If we use the inequality of arithmetic and geometric means, we get
$$A\geq \frac{3}{\sqrt[3]{xyz}}=\frac{3}{\sqrt[3]{x+y+z}}.$$
 [/INST] From our constraint $x + y + z = xyz$, we obtain that $z = \frac{x+y}{xy-1}$, plugging that into
$$\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}$$
we obtain
$$\frac{x}{y^2} + \frac{y(xy-1)^2}{(x+y)^2} + \frac{x+y}{x^2 (xy-1)}$$
Call this $f(x,y)$. We can then find a minimum value by doing the second derivative test. First, we need to find the critical points by setting
$$f(x,y) = \frac{x}{y^2} + \frac{y(xy-1)^2}{(x+y)^2} + \frac{x+y}{x^2 (xy-1)} = 0$$
We would obtain some ordered pair $(a, b)$ (there may be more than one set of critical points - there may be infinitely many even!). From this we calculate the Hessian,
$$H = \det \begin{pmatrix} f_{xx}(a,b) & f_{xy}(a,b) \\ f_{yx}(a,b) & f_{yy}(a,b)\end{pmatrix}$$
For a minimum we want $H > 0$ and $f_{xx}(a,b) > 0$, once we find that culprit, we have found our minimum.
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