01/08/2022

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Largest value of $\sqrt[3]{xyz} – \sqrt[3]{x} – \sqrt[3]{y} – \sqrt[3]{z}$ when $x,y,z \in \mathbb{N}$

Given $x,y,z \in \mathbb{N}$ let
$$f(x,y,z)=\sqrt[3]{xyz} – \sqrt[3]{x} – \sqrt[3]{y} – \sqrt[3]{z}$$
Calculate $$\max \{f(x,y,z) \mid x,y,z \in \mathbb{N} \}$$
My initial idea was to use Lagrange Multipliers and try to maximize a linear form with constraints.
$$\max \Big\{ \underbrace{\frac{y}{x+y+z}}_{a} – \sqrt[3]{xyz} – \underbrace{\sqrt[3]{x}}_{b} – \sqrt[3]{y} – \sqrt[3]{z} \; : \; 0\leq a,b \leq 1 \Big\}$$
$$

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