<s>[INST] Inequality Proof $\frac{a}{1-a^2}+\frac{b}{1-b^2}+\frac{c}{1-c^2}\geq \frac{3\sqrt{3}}{2}$ Let $a,b,c\in \mathbb{R}^+$, and $a^2+b^2+c^2=1$, show that:
$$ \frac{a}{1-a^2}+\frac{b}{1-b^2}+\frac{c}{1-c^2}\geq \frac{3\sqrt{3}}{2}$$
 [/INST] Here is a hint:  it is enough to show for $x\in (0, 1)$,
$$f(x) = \left(\frac{x}{1-x^2}-\frac{\sqrt3}2\right)-\frac{3\sqrt3}2\left(x^2-\frac13 \right) \geqslant 0$$
$$\iff \frac{x(\sqrt3x+2)(3x-\sqrt3)^2}{6(1-x^2)} \geqslant 0$$
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