Question

`f(x)=[2x]sin3pix` and `f'(k^+)=lamdakpi(-1)^k`, where [.] denoted greater integer function and `kinN`, then what is the value of `lambda` ?

Answer 1

`lambda = 6`

Explanation:

Considering `d/dx[2x]=0` we have

`(df)/dx = 3pi[2x]cos(3pi x)`

and now if `k in NN` we have `[2k]=2k` so

`3pi 2k cos(3pi k)=lambda k pi(-1)^k` or

`6k pi cos(3k pi)=lambda k pi(-1)^k` and

`cos(3k pi) = (-1)^k` and then

`6k pi(-1)^k = lambda k pi(-1)^k`

and the solution is for `lambda = 6`