Please Help

As we need to find the max value of a product we can use AM-GM Inequality. the inequality states that for any real numbers $x_1, x_2, \ldots, x_n \geq 0$,
$\LARGE\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}$

with equality if and only if $x_1 = x_2 = \cdots = x_n$

Thus using this we write:
$\Large\frac{x^2+3y^2}{2}\ge\sqrt[2]{x^2\cdot3y^2}$

Plugging in values and solving-->
$\sqrt3xy\le9\\ \boxed{xy\le3\sqrt3}$

I guess draw the ellipse