Prove that, if \(a\), \(b\) are real, \[ab\le \left(\frac{a+b}{2}\right)^2,\] and deduce that, if \(a\), \(b\), \(c\), \(d\) are positive, \[abcd\le \left(\frac{a+b+c+d}{4}\right)^4,\] with equality only when all numbers are equal.
By giving \(d\) a suitable value in terms of \(a\), \(b\), \(c\), or otherwise, prove that, if \(a\), \(b\), \(c\) are positive, \[abc\le \left(\frac{a+b+c}{3}\right)^3.\]
