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

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Feb 21, 2018

$\lambda = 6$

#### Explanation:

Considering $\frac{d}{\mathrm{dx}} \left[2 x\right] = 0$ we have

$\frac{\mathrm{df}}{\mathrm{dx}} = 3 \pi \left[2 x\right] \cos \left(3 \pi x\right)$

and now if $k \in \mathbb{N}$ we have $\left[2 k\right] = 2 k$ so

$3 \pi 2 k \cos \left(3 \pi k\right) = \lambda k \pi {\left(- 1\right)}^{k}$ or

$6 k \pi \cos \left(3 k \pi\right) = \lambda k \pi {\left(- 1\right)}^{k}$ and

$\cos \left(3 k \pi\right) = {\left(- 1\right)}^{k}$ and then

$6 k \pi {\left(- 1\right)}^{k} = \lambda k \pi {\left(- 1\right)}^{k}$

and the solution is for $\lambda = 6$

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