tackle an integer

The given expression is $3^{x(3-x)}=3^{3x-x^2}$

$=3^{-(x-\frac{3}{2})^2+\frac{9}{4}}$

$\leq 3^{\frac{9}{4}}\approx 11.844666...$

Hence the required answer is $\boxed {11}$ .

If the expression was $3^x(3-x)$ :

Differentiating with respect to $x$ and equating with zero, we see that the extremum of the expression occurs at $x=2$ . At $x=0$ , the differential coefficient is positive, showing that the expression is increasing at this point. At $x=3$ , the value of the expression is zero. This implies that the extremum is actually a maximum. The maximum value of the expression at $x=2$ is $\boxed 9$ .