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He wondered what the equation of such a perfect roller coaster

would be. As he dreamed that night, he discovered the following equation:
๐ป(๐‘‘) = ๐‘Ž๐‘‘
3 + ๐‘๐‘‘
2 + ๐‘๐‘‘ + ๐‘’ 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.
๐‘–. ๐‘กโ„Ž๐‘’ ๐‘Ÿ๐‘œ๐‘™๐‘™๐‘’๐‘Ÿ ๐‘๐‘œ๐‘Ž๐‘ ๐‘ก๐‘’๐‘Ÿ ๐‘š๐‘ข๐‘ ๐‘ก โ„Ž๐‘Ž๐‘ฃ๐‘’ ๐‘Ž ๐‘™๐‘œ๐‘๐‘Ž๐‘™ ๐‘š๐‘Ž๐‘ฅ๐‘–๐‘š๐‘ข๐‘š ๐‘Ž๐‘ก ๐‘กโ„Ž๐‘’ ๐‘๐‘œ๐‘–๐‘›๐‘ก ๐‘คโ„Ž๐‘’๐‘› ๐‘‘ =1/2
๐‘–๐‘–. ๐‘‡โ„Ž๐‘’ ๐‘Ÿ๐‘œ๐‘™๐‘™๐‘’๐‘Ÿ ๐‘๐‘œ๐‘Ž๐‘ ๐‘ก๐‘’๐‘Ÿ ๐‘š๐‘ข๐‘ ๐‘ก โ„Ž๐‘Ž๐‘ฃ๐‘’ ๐‘Ž ๐‘™๐‘œ๐‘๐‘Ž๐‘™ min ๐‘คโ„Ž๐‘’๐‘› ๐‘‘ = 2 , โ„Ž = 0.5
๐‘–๐‘–๐‘–. ๐‘‡โ„Ž๐‘’ ๐‘Ÿ๐‘œ๐‘™๐‘™๐‘’๐‘Ÿ ๐‘๐‘œ๐‘Ž๐‘ ๐‘ก๐‘’๐‘Ÿ ๐‘š๐‘ข๐‘ ๐‘ก โ„Ž๐‘Ž๐‘ฃ๐‘’ ๐‘Ž ๐‘๐‘œ๐‘–๐‘›๐‘ก ๐‘œ๐‘“ ๐‘–๐‘›๐‘“๐‘™๐‘’๐‘๐‘ก๐‘–๐‘œ๐‘› ๐‘Ž๐‘ก ๐‘กโ„Ž๐‘’ ๐‘๐‘œ๐‘–๐‘›๐‘ก ๐‘คโ„Ž๐‘’๐‘› ๐‘‘ = 1.25
๐‘–๐‘ฃ. ๐‘‡โ„Ž๐‘’ ๐‘Ÿ๐‘œ๐‘™๐‘™๐‘’๐‘Ÿ ๐‘๐‘œ๐‘Ž๐‘ ๐‘ก๐‘’๐‘Ÿ ๐‘ ๐‘ก๐‘Ž๐‘Ÿ๐‘ก๐‘  ๐‘Ž๐‘ก ๐‘Ž โ„Ž๐‘’๐‘–๐‘”โ„Ž๐‘ก ๐‘œ๐‘“ 1 ๐‘š ๐‘Ž๐‘๐‘œ๐‘ฃ๐‘’ ๐‘กโ„Ž๐‘’ ๐‘”๐‘Ÿ๐‘œ๐‘ข๐‘›๐‘‘
Determine any such equation satisfying the above using methods of calculus only! Your
solutions must be neat and organized.

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"