Roller Coaster Track G-Force Curvature Transitions (Entertainment Engineering)

In themed entertainment engineering and roller coaster track design, ride dynamics must deliver thrilling accelerations while strictly respecting human physiological safety tolerances. Vertical track curvature dictates centripetal accelerations: concave upward valleys generate high positive normal forces (compression Gs into the seat), while concave downward hill crests produce negative normal forces (airtime Gs). Abrupt jumps in curvature generate dangerous lateral jerk forces that cause cervical strain.

An entertainment engineer at a Las Vegas Strip resort is analyzing a high-speed drop-and-camelback track profile. The track elevation $y(x)$ (in meters) as a function of horizontal position $x$ (in tens of meters, for $x \in [0, 4]$) is modeled by the polynomial function:

1. Determine the track elevation at the beginning of the drop element $x = 0$ ($y$-intercept).
2. Calculate the first derivative $y'(x)$ (the track grade/slope), locate all critical points on $[0, 4]$, and classify them using the First Derivative Test.
3. Compute the second derivative $y''(x)$, determine all intervals of concavity, and identify all points of inflection.
4. Construct an Information Consolidation Table synthesizing track coordinates, slopes, and curvature states.
5. In entertainment safety standards, an inflection point ($y''(x) = 0$) represents zero vertical centripetal acceleration ($a_n = \kappa v^2 = 0$), where passengers experience pure $1.0\text{ G}$ gravitational free-fall. Explain how tracking these inflection points guides the placement of magnetic trim brakes.