He wondered what the equation of such a perfect roller coaster

Question

would be. As he dreamed that night, he discovered the following equation:
𝐻(𝑑) = 𝑎𝑑
3 + 𝑏𝑑
2 + 𝑐𝑑 + 𝑒 would work, where H(d) represents the height above the
ground, and d represents the horizontal distance the coaster has travelled.
In order to make his perfect roller coaster, he wanted the following to be true.
𝑖. 𝑡ℎ𝑒 𝑟𝑜𝑙𝑙𝑒𝑟 𝑐𝑜𝑎𝑠𝑡𝑒𝑟 𝑚𝑢𝑠𝑡 ℎ𝑎𝑣𝑒 𝑎 𝑙𝑜𝑐𝑎𝑙 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑤ℎ𝑒𝑛 𝑑 =1/2
𝑖𝑖. 𝑇ℎ𝑒 𝑟𝑜𝑙𝑙𝑒𝑟 𝑐𝑜𝑎𝑠𝑡𝑒𝑟 𝑚𝑢𝑠𝑡 ℎ𝑎𝑣𝑒 𝑎 𝑙𝑜𝑐𝑎𝑙 min 𝑤ℎ𝑒𝑛 𝑑 = 2 , ℎ = 0.5
𝑖𝑖𝑖. 𝑇ℎ𝑒 𝑟𝑜𝑙𝑙𝑒𝑟 𝑐𝑜𝑎𝑠𝑡𝑒𝑟 𝑚𝑢𝑠𝑡 ℎ𝑎𝑣𝑒 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑤ℎ𝑒𝑛 𝑑 = 1.25
𝑖𝑣. 𝑇ℎ𝑒 𝑟𝑜𝑙𝑙𝑒𝑟 𝑐𝑜𝑎𝑠𝑡𝑒𝑟 𝑠𝑡𝑎𝑟𝑡𝑠 𝑎𝑡 𝑎 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 1 𝑚 𝑎𝑏𝑜𝑣𝑒 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑
Determine any such equation satisfying the above using methods of calculus only! Your
solutions must be neat and organized.

Answer 1

you know where y'=0 and y" = 0
so write those equations to solve for a,b,c
also, you know e = 1.25
Do that and you will wind up with
y = 4x^3-15x^2+12x+1

This same problem has been posted before. See Similar Questions below, under "calculus"