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

## Question

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

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

3. Show that $a+b+c$ is a factor of $a^3+b^3+c^3-3abc$ and find the other factor of this expression.

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