Let α > 0, β > 0 be such that α ^ 3 + β ^ 2 = 4 . If the maximum...

${T}_{r + 1} = Cr 10 {\left(\alphax\dfrac{1}{9}\right)}^{10 - r}{\left(\beta{x}^{- \dfrac{1}{6}}\right)}^{r}$

${T}_{r + 1} = Cr 10 {\alpha}^{10 - r}{\beta}^{r}{\left(x\right)}^{\dfrac{10 - r}{9} - \dfrac{r}{6}}$

Term independent of $x ⇒$Power of $x = 0$

$\dfrac{10 - r}{9} - \dfrac{r}{6} = 0 ⇒ r = 4$

${T}_{5} = C4 10 {\alpha}^{6}{\beta}^{4}$

Now Let ${\alpha}^{3}, {\beta}^{2}$ are two numbers. Clearly both are positive.

$A . M . \ge G . M .$

$⇒ \dfrac{{\alpha}^{3} + {\beta}^{2}}{2} \ge {\left({\alpha}^{3}{\beta}^{2}\right)}^{1 / 2}$

$⇒ {\alpha}^{3}{\beta}^{2} \le 4$

$⇒ {\alpha}^{6}{\beta}^{4} \le 16$

$⇒ \dfrac{{T}_{5}}{10{C}_{4}} \le 16$

$⇒ {T}_{5} \le 16{.}^{10}{C}_{4}$

$⇒ {T}_{5max} = 16{\times}^{10}{C}_{4} = 10k$

$⇒ 10k = 3360$

$⇒ k = 336$