Suppose you run k$-$means using k $=3$ and k $=5.$ You find that the cost function is much

higher for k $=5$ than k $=3$ at convergence. What can you conclude?

$($a$)$ This is mathematically impossible. There must be a bug in the code.

$($b$)$ k$-$means got stuck in a bad local minimum when k $=5.$ You should try re$-$running k$-$means

for k $=5$ with a different initialization.

$($c$)$ k$-$means got lucky when k $=3.$ You should try re$-$running k$-$means for k $=3$ with different

initializations, until it performs no better than k $=5.$