If the function $f(x)=\left[\frac{(x-2)^3}{a}\right] \sin (x-2)+a \cos (x-2)$, [.] denotes the greatest integer function, is continuous and differentiable in (4, 6), then

$a \in[64, \infty)$

We have,

$x \in(4,6)$

$\Rightarrow 4<x<6$

$\Rightarrow 2<x-2<4$

$\Rightarrow 8<(x-2)^3<64 \Rightarrow \frac{8}{a}<\frac{(x-2)^3}{a}<\frac{64}{a}, a>0$

For f(x) to be continuous and differentiable in $(4,6),\left[\frac{(x-2)^3}{a}\right]$ must attain a constant value for all $x \in(4,6)$

Clearly, this is possible only when $a \geq 64$

In that case, we have

$f(x)=a \cos (x-2)$ which is continuous and differentiable.

Hence, $a \in[64, \infty)$