To solve this problem, we need to analyze the behavior of the first and second derivatives of the function \(f(x) = \sin(x)\) at the points P, Q, and R in the given plot.

Let's break down the problem step by step:

1. Consider the first and second derivatives of \(f(x) = \sin(x)\):
   • The first derivative is \(\frac{df}{dx} = \cos(x)\).
   • The second derivative is \(\frac{d^2f}{dx^2} = -\sin(x)\).
2. Evaluate the derivatives at point P (maximum point):
   • At a maximum point, the first derivative \(\frac{df}{dx} = 0\) (since the slope is horizontal).
   • The second derivative is negative, \(\frac{d^2f}{dx^2} < 0\), indicating a local maximum. This is consistent with option (C).
3. Evaluate the derivatives at point Q (inflection point):
   • At point Q, the second derivative \(\frac{d^2f}{dx^2} = -\sin(x) = 0\), indicating an inflection point where the concavity changes. This matches option (C).
4. Evaluate the derivatives at point R (minimum point):
   • At a minimum point, the first derivative \(\frac{df}{dx} = 0\) (slope is again horizontal).
   • The second derivative is positive, \(\frac{d^2f}{dx^2} > 0\), indicating a local minimum. This aligns with option (C).

Based on the analysis above, we conclude that the correct answer is option (C), which states:

\(\frac{d^2f}{dx^2}|_P < 0; \frac{d^2f}{dx^2}|_Q = 0; \frac{d^2f}{dx^2}|_R > 0\)