Solution
= Solution
Using $\max\{X^2,Y^2\}=(X^2+Y^2+|X^2-Y^2|)/2$ and <Cauchy-Schwarz inequality>,
$$
\mathbb E\max\{X^2,Y^2\}
\leq1+\frac12\sqrt{\mathbb E(X-Y)^2\,\mathbb E(X+Y)^2}
=\boxed{1+\sqrt{1-\rho^2}}.
$$
= Solution
Using $\max\{X^2,Y^2\}=(X^2+Y^2+|X^2-Y^2|)/2$ and <Cauchy-Schwarz inequality>,
$$
\mathbb E\max\{X^2,Y^2\}
\leq1+\frac12\sqrt{\mathbb E(X-Y)^2\,\mathbb E(X+Y)^2}
=\boxed{1+\sqrt{1-\rho^2}}.
$$