Question 59
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?
$$
z = (2x - y)^2 - 2y^2 - 3y.
$$
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?