I was wrong: you can ignore LaGrangian multipliers. You cannot use calculus because your functions are not differentiable. They are discrete piecewise functions.
Your data are not sufficient to determine anything. What you need to know is the distribution of cars demanded per day, and then minimize expected cost. I am going to assume that H is the highest required on any day and L is the lowest required every day, and that you know the probability distribution of cars demanded.
[MATH]c(r,\ d) = \text {cost per day if r cars are reserved and d are demanded.}[/MATH]
[MATH]c(r, d) = \dfrac{20r}{3} \text { if } d \le r \text { and }
c(r,\ d) = \dfrac{20r}{3} + 30d - 30r.[/MATH]
Buy that? On my assumptions, d will be an integer from L to H rather than a continuous variable.
[MATH]p(d) = \text {probability that d will be demanded on a day.}[/MATH]
[MATH]e(r) = \text {expected daily cost if r reserved} = \sum_{d=L}^H p(d) * c(r,\ d).[/MATH]
It is obvious that r < L is higher cost than r [MATH]\ge[/MATH] L.
Now pick the lowest expected cost.
If you have hundreds of data points, this is easy to do with a spread sheet.
If you have hundreds of thousands, you will need to get more sophisticated.
