(7) Let $x, y \in[0,1]$, the maximum value of the function $f(x, y)=x \sqrt{1-y}+y \sqrt{1-x}$ is $\qquad$ .
71 Prompt: Since $x, y \in [0,1]$, then $x \leqslant \sqrt{x}, y \leqslant \sqrt{y}$, let
$$
x=\sin ^{2} \alpha, y=\sin ^{2} \beta, \alpha, \beta \in\left[0, \frac{\pi}{2}\right],
$$

thus
$$
\begin{aligned}
f(x, y) & =x \sqrt{1-y}+y \sqrt{1-x} \\
& \leqslant \sqrt{x(1-y)}+\sqrt{y(1-x)} \\
& =\sin \alpha \cos \beta+\cos \alpha \sin \beta \\
& =\sin (\alpha+\beta) \\
& \leqslant 1,
\end{aligned}
$$

equality holds if and only if $\alpha+\beta=\frac{\pi}{2}$, and $x=\sqrt{x}, y=\sqrt{y}$, at this time $\{x, y\}=$ $\{0,1\}$.