# HMMT - Februrary 2015 - GUTS!

Calculus Level 4

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?

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