Correct Option: $C$.
$f(x) = R*\sin(\frac{\pi * x}{2}) + S$
$f'(x) = R*\cos(\frac{\pi*x}{2})*\frac{\pi}{2}$
$f'(\frac{1}{2}) = R*\cos(\frac{\pi}{4})*\frac{\pi}{2} = \sqrt{2}$
$R = \frac{\sqrt{2}*\sqrt{2}*2}{\pi} = \frac{4}{\pi}$
$f(x) = \frac{4}{\pi}*\sin(\frac{\pi*x}{2}) + S$
$\int_{0}^{1} f(x)*dx = \int_{0}^{1}(\frac{4}{\pi}*\sin(\frac{\pi*x}{2}) + S)*dx = \frac{2*R}{\pi} = \frac{8}{\pi^2}$
$\frac{4}{\pi}\int_{0}^{1}\sin(\frac{\pi*x}{2})*dx + \int_{0}^{1}S*dx = \frac{8}{\pi^2}$
$\frac{4}{\pi} \left[-\cos(\frac{\pi*x}{2})*\frac{2}{\pi} \right ]_0^1 + S \left[x \right ]_0^1 = \frac{8}{\pi^2}$
$\frac{8}{\pi^2} \left[-0+1 \right ] + S = \frac{8}{\pi^2}$
$S = 0.$