HMMT 二月 2015 · 冲刺赛 · 第 4 题

英文解析

4. [ 4 ] Consider the function z ( x, y ) describing the paraboloid
   2 2
   z = (2 x − y ) − 2 y − 3 y.
   Archimedes and Brahmagupta are playing a game. Archimedes first chooses x . Afterwards, Brahmagupta chooses y . Archimedes wishes to minimize z while Brahmagupta wishes to maximize z .
   Assuming that Brahmagupta will play optimally, what value of x should Archimedes choose?
   Answer: − Viewing x as a constant and completing the square, we find that 3
   2 2 28
   z = 4 x − 4 xy + y − 2 y − 3 y
   2 2 = − y − (4 x + 3) y + 4 x
   ( ) ( )
   2 2
   4 x + 3 4 x + 3 = − y + + + 4 x .2
   2 2
   4 x +3
   Bhramagupta wishes to maximize z , so regardless of the value of x , he will pick y = − . Theexpression for z then simplifies to 2
   z = 8 x + 6 x + .29
   Guts 4
   Archimedes knows this and will therefore pick x to minimize the above expression. By completing the square, we find that x = − minimizes z .3
   Alternatively, note that z is convex in x and concave in y , so we can use the minimax theorem to switch 8
   the order of moves. If Archimedes goes second, he will set x = to minimize z , so Brahmagupta willy
   2 3 32
   maximize − 2 y − 3 y by setting y = − . Thus Archimedes should pick x = − , as above.
   4 8