September 22nd, 2019, 4:32 pm
We can split the problem. The firs part, just use the arithmetic and geomtric mean inequality ([$] (x_1 + x_2 + x_3) \geq 3 \sqrt[3] {x_1 x_2 x_3} [$])
with [$] x_1 = \frac{x^2}{y z}, x_2= \frac{y^2}{x z}, x_3 = \frac{x^2}{x y} [$] then use the cubic root of the inequality to obtain the first part.
For the second part, find the minimum of the (positive) function [$] f(x, y, z) = \frac{x^2 + y^2 + z^2}{x y + y z + z x} [$] by zeroing the gradient.
We find the solution as [$] (x, y, z) = t (1, 1, 1) [$] with a minimum 1.