American calls and a discrete dividend

A stock pays a single known dividend $D$ at time $t_D<T$ (the ex-dividend date). Interest rate $r>0$. (a) Show that early exercise of an American call is never optimal at any time other than immediately before $t_D$. (b) Show that if $D\le K\big(1-e^{-r(T-t_D)}\big)$, early exercise is never optimal at all. (c) A stock option has $K=\text{₹}1{,}000$, a dividend of ₹40 going ex 10 days before expiry, and $r=6\%$. Is early exercise ruled out by (b)? Is it necessarily optimal? Why is this question irrelevant for NSE stock options?