Question:- Let $a,b$ and $c$ be integers, not all equal to $0$. Show that $$\frac{1}{4a^2+3b^2+2c^2}\leq \left|\sqrt[3]{4}\,a+\sqrt[3]{2}\,b+c\right|.$$
This problem was proposed in a canadian journal. The presented solution is very hard to grasp. Is there a better way to prove this (may be using calculus)?
Essentially the same proof, without the error of assuming $a,b,c>0$ and motivating how we come up with the first note:
The algebraic integer $\alpha_0=\sqrt[3]4a+\sqrt[3]2b+c$ has conjugates $\alpha_1=\zeta^2\sqrt[3]4a+\zeta\sqrt[3]2b+c$ and $\alpha_2=\zeta^4\sqrt[3]4a+\zeta^2\sqrt[3]2b+c$, $\zeta$ a primitive cube root of unity.
Since $\alpha_0\neq 0$ for all $(a,b,c)\in\mathbb{Z}^3-\{(0,0,0)\}$, the complex conjugates $\alpha_1$ and $\alpha_2$ are nonzero, hence the real numbers $\alpha_0$ and its number-theoretic norm $$ N_{\mathbb{Q}[\sqrt[3]{2}]/\mathbb{Q}}(\alpha_0)=\alpha_0\alpha_1\alpha_2=4a^3+2b^3+c^3-6abc $$ have the same sign. Since $\alpha_0\neq 0$, we have $\lvert N_{\mathbb{Q}[\sqrt[3]{2}]/\mathbb{Q}}(\alpha_0)\rvert\geq 1$. So it remains to prove $$ \alpha_1\alpha_2\leq 4a^2+3b^2+2c^2. $$ Note that $\alpha_1\alpha_2=2\sqrt[3]2a^2+\dots$ is automatically positive, so the $-2\sqrt[3]2a^2-\dots\leq 4a^2+3b^2+2c^2$ part of the given proof is never required.