Question

I know this question, has been asked, but I'm confused.

6a(5/4) + 2b = 0
3a*2^2 + 2b*2 + c = 0
3a(1/2)^2 + 2b(1/2) + c = 0

how do these three systems of equations produce H(d) = 4d^3 - 15d^2 + 12d + 1

The 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 keep on getting zeros a,b,c

Answer 1

come on. You are taking calculus, and you can't do Algebra I?
You have these equations
6a(5/4) + 2b = 0
3a*2^2 + 2b*2 + c = 0
3a(1/2)^2 + 2b(1/2) + c = 0
All those fractions are confusing, so try rewriting them as
15a+4b = 0
12a+4b+c = 0
3a+4b+4c = 0
Clearly, if you eliminate b, you now get
15a+4b = 0
9a-3c = 0
which gives you c = 3a
Using that, you get
5c+4b = 0
so b = -5/4 c = -15/4 a
So, pick any convenient value for a, say, a=4, and now you get
a = 4
b = -15
c = 12
So the solution given is not unique, but it is the simplest to come up with.