If the function `f(x)=[((x-2)^3)/a]sin(x-2)+acos(x-2),[dot]` denotes the greatest integer function, is continuous in `[4,6],` then find the values of `adot`

Options

A. `a in [8,64)`
B. `a in [0,8)`
C. `a in [64,oo)`
D. none of these

Answer

C

Explanation

We have
`x in (4,6)`
`Rightarrow 4 lt x lt 6`
`Rightarrow 2 lt x -2 lt 4`
`Rightarrow 8 lt (x-2)^(3) lt 64 Rightarrow (8)/(a) lt ((x-2)^(3))/(a) lt (64)/(a), agt 0`
For f(x) to be continuous and differentiable in `(4,6), [(x-2)^(3)/(a)]`
must attain a constant value for all `x in (4,6)`
Clearly, this is possible only when `a ge 64`
In that case, we have
`f(x)=a cos (x-2)` which is continuous and differentiable.
Hence, `a in [64,oo)`