If the function f(x)=(x−2)3asin⁡(x−2)+acos⁡(x−2) ,[.] denotes the greatest integer function, is continuous and differentiable in (4, 6), then the least value of a is

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

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### Solution:

We have

For  to be continuous and differentiable in   $\left[\frac{\left(x-2{\right)}^{3}}{a}\right]$

must attain a constant value for all

Clearly, this is possible only when $a\ge 64.$

In that case, we have

$f\left(x\right)=a\mathrm{cos}\left(x-2\right)$ which is continuous and differentiable.

Hence, the least value of $a$ is $64$

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