k means clustering when k = n I want to find the minimum value of the objective function if we set K equal to the number of samples.

I know the objective function is $J=\Sigma_{n=1}^{N}\Sigma_{k=1}^{K}r_{nk}||x_n-\mu_k||^2$
And we take the derivative to get $\mu_k=\frac{\Sigma_{n=1}^{N}r_{nk}x_n}{\Sigma_{n=1}^{N}r_{nk}}$

However, when plugging that back in and set K=N, it seems unable to be simplified:
$J=\Sigma_{n=1}^{N}\Sigma_{k=1}^{N}r_{nk}||x_n-\frac{\Sigma_{n=1}^{N}r_{nk}x_n}{\Sigma_{n=1}^{N}r_{nk}}||^2$

Any help?
Well, when k = n, the obvious global minimum occurs where $x_n = \mu_k$ for all cluster centers. This falls out from your derivate because $r_{nk} = 1$ iff $n = k$.