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.

Try and apply $x + y + z \geq 3 \sqrt[3]{xyz}$ to each factor in the left-hand side of the inequality to prove.
