Problem
ALG-B2-M05-P004 Product with fixed sum
#4
★★☆☆☆ Level 2 of 5
Let \(a,b,c>0\) and \(a+b+c=6\). Prove that \(abc\le8\).
Hint. Apply Jensen to \(\ln x\).
Since \(\ln x\) is concave, \[\frac{\ln a+\ln b+\ln c}{3}\le\ln\frac{a+b+c}{3}=\ln2.\] Hence \(\ln(abc)\le3\ln2=\ln8\), so \(abc\le8\).
Shows the standard transition from a product to a sum of logarithms.