Problem
ALG-B2-M05-P010 Maximum product
#10
★★★☆☆ Level 3 of 5
Let \(a,b,c>0\) and \(a+b+c=12\). Find the maximum possible value of \(abc\).
Hint. Apply Jensen to \(\ln x\), then return from logarithms to the product.
Since \(\ln x\) is concave, \[\frac{\ln a+\ln b+\ln c}{3}\le\ln\frac{a+b+c}{3}=\ln4.\] Therefore \(\ln(abc)\le\ln64\), so \(abc\le64\). Equality holds at \(a=b=c=4\).
This problem reinforces the chain product - logarithm - concavity.