Problem
ALG-B2-M09-P008 Shifted product
#8
★★★☆☆ Level 3 of 5
Let \(a,b,c\ge0\), \(a+b+c=3\). Prove \[(1+a)(1+b)(1+c)\le8.\]
Hint. Take logarithms and apply Jensen to \(\ln(1+x)\).
The function \(\ln(1+x)\) is concave for \(x\ge0\). Hence \(\frac13\sum\ln(1+a)\le\ln(1+(a+b+c)/3)=\ln2\). Therefore \(\ln\prod(1+a)\le\ln8\), and the product is at most \(8\).
The constraint turns a product problem into a concave logarithm problem.