- 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?
- 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?
- 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?
