Proving an equality using given ones; no need for differentiation Prove that $(\frac ab+\frac bc+\frac ca)(\frac ba+\frac cb+\frac ac)\geqslant9$
The formulas given were
$$\frac{a+b} {2}\geqslant\sqrt {ab}$$
$$a^2+b^2\geqslant2ab$$
$$\frac{a+b+c} {3}\geqslant\root3\of{abc}$$
$$a^3+b^3+c^3\geqslant3abc$$
for all $a\gt0, b\gt0, c\gt0.$
I couldn't think of any way to do this; please help.

Use that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3\sqrt[3]{\frac{a}{b}\frac{b}{c}\frac{c}{a}}=3$$
and $$\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\geq 3\sqrt[3]{\frac{b}{a}\frac{c}{b}\frac{a}{c}}=3$$
