Question:

\(\text{ If } f(x) = \sin x + \frac{1}{2} \cos 2x \text{ in } \left[ 0, \frac{\pi}{2} \right], \text{ then:}\)
(A) \(f'(x) = \cos x - \sin 2x\)
(B)The critical points of the function are \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\)
(C) The minimum value of the function is 2
(D) The maximum value of the function is \(\frac{3}{4}\)

Updated On: Nov 15, 2024
  • (A), (B), and (D) only
  • (A), (B), and (C) only
  • (B), (C), and (D) only
  • (A), (C), and (D) only
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The Correct Option is A

Solution and Explanation

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