Question

The given function H(d) = ad^3 + bd^2 + cd + e represents a roller coaster, where H(d) represents the height above the ground, and d represents the horizontal distance the roller coaster has travelled.

Make it so the following parameters are true:

1) The roller coaster must have a local maximum at the point when d = (1)/2
2) The roller coaster must have a local minimum when d = 2 , h = 0.5
3) The roller coaster must have a point of inflection at the point when d = 1.25
4) The roller coaster starts at a height of 1 m above the ground.

I've tried each part separately, and I can get it to work but as soon as I put the parts together it seems to fall apart, I'm confused

Answer 1

so, just what did you do? Too bad you don't show anything we can work with.
H(d) = ad^3 + bd^2 + cd + e
H'(d) = 3ad^2 + 2bd + c
H"(d) = 6ad + 2b

So, we need
H"(1/2) < 0
H"(2) > 0
H(2) = 1/2
H"(5/4) = 0
H(0) = 1
Putting all that into equations, we need to solve
6a(5/4) + 2b = 0
e = 1
3a*2^2 + 2b*2 + c = 0
3a(1/2)^2 + 2b(1/2) + c = 0
or,
15a+4b = 0
12a+4b+c = 0
3a+4b+4c = 0
Solving those, we get
H(d) = 4d^3 - 15d^2 + 12d + 1
H'(d) = 6(2x^2-5x+2)
H"(d) = 6(4x-5)
H has a local maximum at (1/2, 15/4)
H has a local minimum at (2,-3)
H has a point inflection at (5/4, 3/8)
H(0) = 1