T_r+1 = ⁹C_r a^a–r b^r ${\mathrm{x}}^{\frac{3-\mathrm{r}}{2}}$ ⇒   r = 3 (for term independent of x)
$\therefore$ Term independal at x is equal to  ⁹C₃ a⁶ b³     ……(i)
to get the maximum value of a⁶b³ use AM $\ge$ GM

i.e., $\frac{3\left(\frac{{\mathrm{a}}^2}{3}\right)+3\left(\frac{\mathrm{b}}{3}\right)}{6}$ $\ge$ ${\left[\frac{{\mathrm{a}}^6{\mathrm{b}}^3}{3^6}\right]}^{\frac{1}{3}}$
$\therefore$   (a⁶b³)_max = 1
$\therefore$  Answer = ⁹C³