Let be a Poisson distribution. Show that increases as it approaches to , reaching it's max when k is the largest integer less then then decrease as

Hint: Consider .

since the Poisson distribution is defined as:

I was thinking of using derivatives to max out the equation with respect to k, but realized that due to the factorial I can't since it's not continuous.

I'm not sure I'm doing this right, but if I apply the hint I would get:

now I'm not sure what I have to do