Review question

# Can we prove the Arithmetic and Geometric Mean Inequality? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6546

## Suggestion

1. Prove that, if $a$ and $b$ are real, $\frac{1}{2}(a^2+b^2)\geq ab.$

What happens if we bring everything to the left-hand side and try to factorise?

1. Deduce that, if $a$,$b$ and $c$ are real, $a^2+b^2+c^2\geq ab+bc+ca.$

Can we use the first part of the question here?

1. Deduce that, if $p$, $q$ and $r$ are positive, $\frac{1}{3}(p+q+r)\geq (pqr)^{\frac{1}{3}}.$

Could we use earlier parts to help? Would putting $p = a^3, q = b^3, r = c^3$ be more convenient?