Problem
ALG-B2-M05-P017 Product with ones
#17
★★★★★ Level 5 of 5
Let \(x,y,z\ge0\) and \(x+y+z=3\). Prove \[(1+x)(1+y)(1+z)\le8.\]
Hint. Apply Jensen to the function \(\ln(1+t)\).
The function \(\ln(1+t)\) is concave for \(t\ge0\). Thus \[\frac{\ln(1+x)+\ln(1+y)+\ln(1+z)}{3}\le\ln\left(1+\frac{x+y+z}{3}\right)=\ln2.\] Hence the logarithm of the product is at most \(\ln8\), so the product is at most \(8\).
A more hidden function choice: the logarithm is applied to \(1+x\), not to \(x\).