\(- \text{Differentiate } f(x) = \sin x + \frac{1}{2} \cos 2x \text{ to find } f'(x) = \cos x - \sin 2x, \text{ verifying option (A).}\)
\(- \text{Find the critical points by setting } f'(x) = 0, \text{ confirming option (B).}\)
\(- \text{Evaluate the function at critical points to confirm the minimum and maximum values, verifying option (D) and showing that (C) is incorrect.}\)