Transfer of Ideas Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

The AM-GM inequality

\frac{a+b}{2} \ge \sqrt{ab}

was originally about two positive numbers. Transfer the idea to prove: for positive reals

x, y, z

x+y+z \ge 3\sqrt[3]{xyz}

1
Apply AM-GM to two pairs: $x+y \ge 2\sqrt{xy}$ and then apply again to $2\sqrt{xy}$ and $z$ ... (this approach requires care).
2
Alternatively, use the three-variable AM-GM directly: the same idea (arithmetic mean $\ge$ geometric mean) extends to $n$ variables: $\frac{x+y+z}{3} \ge \sqrt[3]{xyz}$ .
3
Multiply both sides by 3: $x+y+z \ge 3\sqrt[3]{xyz}$ . Equality holds when $x=y=z$ .

Answer

x+y+z \ge 3\sqrt[3]{xyz} \text{ for positive reals } x,y,z

The AM-GM idea (mean of values

\ge

geometric mean) transfers from

n=2

to general

n

. Recognising this transfer saves effort: instead of a new proof, we extend a known idea.

About Transfer of Ideas

The ability to recognize that a technique or concept from one area of mathematics applies, possibly in adapted form, to a different area.

The idea of completing the square to solve [formula] transfers to converting [formula] to vertex for

The factorisation [formula] transfers to factoring [formula]. Apply it.

The proof technique 'assume the hypothesis and derive the conclusion' (direct proof) from logic tran