The kinetic energy of gas molecules at a given temperature is given by:
\[ KE = \frac{f}{2}kT \]
where \( f \) is the degrees of freedom and \( kT \) is the thermal energy term.
This is a conceptual fact based on the equipartition theorem, which states that all gases at a given temperature have the same average kinetic energy per molecule.
Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is: