A:

Create a polynomial function that satisfies the following conditions:
has an initial height of 0 m.
reaches a height of 450 m after 1 sec.
Dives below the ground at 3 m.
Comes out of the ground at 6 s
Then goes the ground barely touches it and comes back up at 11 s
Then dives into the ground at 14 s and comes back out a second later - at which point the ride ends.
B.
Using the function you created in part A determine the local maximum and minimum heights of the roller coaster.
Determine the intervals of increase and decrease.
Determine the points of inflection and the intervals where the function is concave up and concave down.

die again and again

To create a polynomial function that satisfies the given conditions, we can work step by step to determine the equation.

Step 1: Initial height of 0 m.
We know that the roller coaster starts at a height of 0 m, so the function will start with a constant term of 0: f(t) = 0.

Step 2: Reaches a height of 450 m after 1 sec.
After 1 second, the height of the roller coaster reaches 450 m. To incorporate this, we need to add a term to the function that grows to 450 m at t = 1. Since it is a polynomial function, we can add a linear term that represents the change in height over time: f(t) = kt.

Substituting t = 1 into the equation, we get k = 450. So, the function becomes f(t) = 450t.

Step 3: Dives below the ground at 3 m.
After reaching a height of 450 m, the roller coaster dives below the ground. This means that between t = 1 and t = 6, there will be a downward trend in the function. To incorporate this, we can add a quadratic term to the function that causes a downward curve. Let's assume the quadratic term to be -at^2, where a is a constant.

Since the roller coaster dives down below the ground at 3 m (f(t) = -3), we can substitute t = 3 into the equation and solve for a:
f(t) = 450t - a(t^2)
-3 = 450(3) - a(3^2)
-3 = 1350 - 9a
9a = 1353
a = 150.33 (rounded to two decimal places)

So, the function becomes f(t) = 450t - 150.33(t^2).

Step 4: Comes out of the ground at 6 s, barely touches the ground at 11 s, and dives into the ground at 14 s.
Between t = 6 and t = 11, the function needs to show an upward trend since the roller coaster comes out of the ground and barely touches it. Finally, between t = 11 and t = 14, there should be another downward trend as the roller coaster dives into the ground again.

We can add a cubic term to account for these changes. Let's assume the cubic term to be bt^3, where b is a constant. We will only use the cubic term between t = 6 and t = 11 and between t = 11 and t = 14.

At t = 11, the height of the roller coaster barely touches the ground, which means f(t) = 0. We can substitute t = 11 into the equation and solve for b:
f(t) = 450t - 150.33(t^2) + bt^3
0 = 450(11) - 150.33(11^2) + b(11^3)
0 = 4950 - 18299.83 + 1331b
1331b = 13349.83
b = 10.02 (rounded to two decimal places)

So, the complete polynomial function is:
f(t) = 450t - 150.33(t^2) + 10.02(t^3)

Now, let's move on to part B to determine the local maximum and minimum heights, intervals of increase and decrease, points of inflection, and intervals of concavity.

To find the local maximum and minimum heights, we need to calculate the critical points of the function by setting its derivative equal to zero and solving for t.

First, let's find the derivative of the function:
f'(t) = 450 - 300.66t + 30.06(t^2)

To find the critical points, set f'(t) equal to zero:
0 = 450 - 300.66t + 30.06(t^2)
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a).

In this case, a = 30.06, b = -300.66, and c = 450.
Using the quadratic formula, we get:
t = (-(-300.66) ± sqrt((-300.66)^2 - 4(30.06)(450))) / (2(30.06))

Simplifying this equation will give us the values of t, which are the critical points where the slope of the function is zero.

Once we find the critical points, we can substitute them back into the original function to determine the corresponding heights.

To determine the intervals of increase and decrease, we look at the sign of the derivative function (f'(t)) in different intervals. If f'(t) is positive, the function is increasing, and if f'(t) is negative, the function is decreasing.

To find the points of inflection and intervals of concavity, we need to find the second derivative (f''(t)) of the function. The points of inflection occur where f''(t) is equal to zero or undefined. The intervals of concavity can be determined by analyzing the sign of f''(t) in different intervals. If f''(t) is positive, the function is concave up, and if f''(t) is negative, the function is concave down.

By following these steps, you can determine the local maximum and minimum heights, intervals of increase and decrease, points of inflection, and intervals of concavity for the roller coaster function you created in part A.