HMMT - Februrary 2015 - GUTS!
Consider the function \( z(x, y) \) describing the paraboloid \( z = (2x - y)^2 -2y ^2 - 3y \).
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?