Question
If the function $$f\left( x \right) = \left[ {\frac{{{{\left( {x - 2} \right)}^3}}}{a}} \right]\sin \left( {x - 2} \right) + a\,\cos \left( {x - 2} \right),\,\left[ . \right]$$ denotes the greatest integer function is continuous and differentiable in $$\left[ {4,\,6} \right]$$ then :
A.
$$a\, \in \,\left[ {8,\,64} \right]$$
B.
$$a\, \in \left( {0,\,8} \right]$$
C.
$$a\, \in \left[ {64,\,\infty } \right)$$
D.
none of these
Answer :
$$a\, \in \left[ {64,\,\infty } \right)$$
Solution :
Since. $$\left[ {{x^3}} \right]$$ is not continuous and differentiable at integral points. So, $$f\left( x \right)$$ is continuous and differentiable in $$\left[ {4,\,6} \right]$$ if $$\left[ {\frac{{{{\left( {x - 2} \right)}^3}}}{a}} \right] = 0\,\, \Rightarrow a \geqslant 64$$