Zadanie 2a, Paper 3, maj 2023, TZ1, Matematyka AA IB HL

A design uses n positive dimensionless parameters x_1,…,x_n. Each parameter consumes one unit of setup resource plus x_j² units of variable resource. The total budget is $n+\sum_{j=1}^n x_j^2=S$. For S>1 and positive integer n<S, let M_n(S) be the largest achievable product x_1⋯x_n, and let P(S) be the largest M_n(S) over all admissible integers n. Initially take n=2 and S=18. You may use AM–GM: for positive z_j, $(\prod z_j)^{1/n}\le(\sum z_j)/n$. For continuous comparison define g(x)>0 by ln g(x) = (x/2)ln(S/x − 1) on 0<x<S.