can you please show all you work to the question below with clean hand writing

1. "e" was very happy as he was going to Canada's Wonderland. He envisioned the perfect roller coaster. One that undulating hills and very thrilling climbs and descents. He wondered what the equation of such a perfect roller coaster would be. As he dreamed that night, he discovered the following equation:

() = ^3 + ^ 2 + + 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.

. = 1/ 2

. min = 2 , = 0.5

. = 1.25

. 1

Determine any such equation satisfying the above using methods of calculus only! Your solutions must be neat and organized.