1230

已知 $m, n$ 均为正数, 求 $2 m^{2} + n^{2} - 2 m n - m + 2 n + 1 + \frac{36}{m} - \frac{36}{m + 1}$ 的最小值.

解

整理得 $原式 = (n - m + 1)^{2} + m (m + 1) + \frac{36}{m (m + 1)} \geq 12,$ 取等条件为 ${\begin{aligned} m (m + 1) = 6, \\ n - m + 1 = 0 \end{aligned} \Rightarrow {\begin{aligned} m = 2, \\ n = 1. \end{aligned}$