Problem
ALG-B2-M05-P023 General power product
#23
★★★★★ Level 5 of 5
Let \(x_1,\ldots,x_n>0\) and \(x_1+\cdots+x_n=n\). Prove \[\prod_{i=1}^n x_i^{x_i}\ge1.\]
Hint. Take the logarithm of the product and apply Jensen to \(t\ln t\).
Since \(f(t)=t\ln t\) is convex for \(t>0\), Jensen gives \[\frac1n\sum_{i=1}^n x_i\ln x_i\ge f\left(\frac{x_1+\cdots+x_n}{n}\right)=f(1)=0.\] Hence \(\ln\prod x_i^{x_i}=\sum x_i\ln x_i\ge0\). Therefore \(\prod x_i^{x_i}\ge1\).
A strong problem on transferring the three-variable idea to a general form.