Expanding $z,$ we get
\begin{align*}
z &= 4x^2 - 4xy + y^2 - 2y^2 - 3y \\
&= -y^2 - (4x + 3) y + 4x^2.
\end{align*}After Archimedes chooses $x,$ Brahmagupta will choose
\[y = -\frac{4x + 3}{2}\]in order to maximize $z.$  Then
\begin{align*}
z &= -\left( -\frac{4x + 3}{2} \right)^2 - (4x + 3) \left( -\frac{4x + 3}{2} \right)^2 + 4x^2 \\
&= 8x^2 + 6x + \frac{9}{4}.
\end{align*}To minimize this expression, Archimedes should choose $x = -\frac{6}{16} = \boxed{-\frac{3}{8}}.$
Question:
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?
Answer:
Let's think step by step.

user
-\frac{3}{8}
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?
test