Exercise 49

Find the maximum value of \(f(x_1, x_2, ..., x_n) = \sqrt[n]{x_1 x_2 ... x_n}\) given that \(x_1, x_2, ..., x_n\) are positive numbers and \(\sum_{i=1}^n x_i = c\), where c is a constant. Deduce from part (a) that if \(x_1, x_2, ..., x_n\) are positive numbers, then \(\sqrt[n]{x_1 x_2 ... x_n} \le \frac{x_1 + x_2 + ... + x_n}{n}\) This inequality says that the geometric mean of n numbers is no larger than the arithmetic mean of the numbers. Under what circumstances are these two means equal?