Question

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

Answer 1

$f\left(x\right)=\left[\frac{(x-2{)}^3}{a}\right]\sin (x-2)+a\cos (x-2)$
$f\left(x\right)=\left[\frac{(x-2{)}^3}{a}\right]=0;\forall x\in [4,6]$
put $x=4$
$\left[\right(4-2{)}^3]=8$
put $x=6$
$\left[\right(6-2{)}^3]=4^3=64$
$\left[\right(x-2{)}^3]\in [8,64]$
$\frac{(x-2{)}^3}{a}<1;a>64$