Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 19 / c / Solution

Because all variables are positive, the product constraint is equivalent to ${\sum }_i\ln x_i=\ln a$. The Lagrange multiplier equations forare $1-\lambda /x_i=0$. Thus every $x_i$ equals $\lambda$, and the constraint requires $\lambda =a^{1/n}$. The candidate minimum is thereforeFor a global justification, put $y_i=\ln x_i$. The objective becomes the strictly convex function ${\sum }_i{e}^{y_i}$ on the affine constraint ${\sum }_i{y}_i=\ln a$. A stationary point of this restricted convex function is its unique global minimum, establishing the claim without presupposing the desired inequality.

Finally take $a={\prod }_i{x}_i$ and divide the minimum bound by $n$. This is the arithmetic-geometric mean inequality:with equality exactly when all the positive variables are equal.