Chose the right regression analysis In R I have data where head(data) gives

day   promotion   profit   new_users
1           105    45662          33
2            12    40662          13
3            44    46800          20
4           203    54102          46


Now day is simply the day (and is in order). promotion is simply the promotion-value for the day, the profit is the profit that day and new_users is the number of new users that day.

I want investigate the relationships between promotion to profit and new_users. We see a clear positive correlation between promotion and profit, and there is also a positive correlation between promotion and new_users.
In R I simply test correlation 

cor.test(data$promotion, data$profit, method="kendall", alternative="greater" )
cor.test(data$promotion, data$new_users, method="kendall", alternative="greater")


which both gives a low p-value, ie we have a positive correlation. 

I want to find a point where where the increase of promotion don't increase profit or new_users that must, ie a sweet spot. 

Here is 2 plots and the R code for these

plot(data$promotion, data$profit, col="brown")
plot(data$promotion, data$new_users)






How should this be done?

My thoughts where to make a regression model.
For the first one "promotion vs. new_users" one could use a poisons model because it's a count-process, so a model like this would be a good chose?

glm(formula= data$new_users ~ data$promotion, family="poisson", data=data)


Next what regression model should one chose for the next one. Is it fair to say that this regression model is a good chose ? (I use sqrt command)

glm(formula=data$profit ~ sqrt(data$promotion) , data=data)


Or maybe it's not even necessary to use a regression model at all to find a sweet spot?

Thanks. 

I have now looked at 'good' new users. For each day we have a promotion value and we have a count value which is the number of new good users. This plot shows us the number of good new users we get for a promotion for each day. For example for promotion value 90 we have a day where we got 8 new good users and a day where we got 14 new good users.  

What would be the right approach to find a sweet spot for the use of promotion ?
Since I cannot comment because I don't have enough reputation, I will post this as an answer.

If your goal is to "find a point where the increase of promotion don't increase profit or new_users", I won't do a simple regression, since the regression will tell you that if you do more promotions, you will always inscrease the profits. I would say that, in reality, the relationship between promotions and profits or new users is not linear. Because the number of new users is limited and the promotions are not.

A better model is to say that there is a optimal promotion that will give you the best increase of profits and new users.

(if you have a real business to optimise, I would introduce the Customer Lifetime Value of new users. Because generally, the new users you get when doing huge promotions will not come back...)