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 \(\frac{3}{4}\)

• A. (A), (B), and (D) only
• B. (A), (B), and (C) only
• C. (B), (C), and (D) only
• D. (A), (C), and (D) only

Hint:

To analyze the critical points of a function, always start by differentiating the function and setting the derivative equal to zero. After finding the critical points, evaluate the function at these points to determine whether they correspond to maximum, minimum, or saddle points. Additionally, always verify each option thoroughly to ensure the correct interpretation of the function’s behavior.

Answer 1

The Correct Option is A

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

Answer 2

Step 1: Differentiate the function \( f(x) = \sin x + \frac{1}{2} \cos 2x \):

We apply the basic differentiation rules to differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx} \left( \sin x + \frac{1}{2} \cos 2x \right) \]

By differentiating each term, we get:

\[ f'(x) = \cos x - \sin 2x \]

This confirms option (A).

Step 2: Find the critical points:

Critical points occur where the derivative is equal to zero. So we solve \( f'(x) = 0 \):

\[ \cos x - \sin 2x = 0 \]

Solving this equation for \( x \), we can find the critical points, which confirms option (B).

Step 3: Evaluate the function at the critical points:

After finding the critical points, we evaluate the original function \( f(x) = \sin x + \frac{1}{2} \cos 2x \) at these points. This allows us to determine if the critical points correspond to a minimum or maximum, confirming option (D) and showing that option (C) is incorrect.