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(x)=(x2)3asin(x2)+acos(x2) ,[.] 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

    x(4,6) 4<x<6

     2<x2<4 8<(x2)3<648a<(x2)3a<64a, a>0 

    For  f (x) to be continuous and differentiable in (4, 6)  (x2)3a

    must attain a constant value for all x ( 4, 6)

    Clearly, this is possible only when a64.

     In that case, we have

    f(x)=acos(x2) which is continuous and differentiable.  

    Hence, the least value of a is 64

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