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The function \( f(x) = \sin x + \frac{1}{2\sqrt{2}} \cos 2x \) has the following first derivative:

\[
f'(x) = \cos x - \frac{\sqrt{2}}{2} \sin 2x
\]

To find the extreme values, we solve \( f'(x) = 0 \) for critical points in the interval \( (0, 2\pi) \). The critical points are approximately:

\[
x = 0.7854, 1.5708, 2.3562
\]

These correspond to \( \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4} \).

Next, you can determine if these are maxima, minima, or points of inflection by examining the second derivative or evaluating the function at these points.

Would you like details on these critical points or further analysis?

Here are some related questions to explore:

1. What is the second derivative of the function to classify the critical points?
2. How do we determine whether these critical points correspond to local maxima or minima?
3. What is the behavior of the function at the boundaries of the interval \( (0, 2\pi) \)?
4. How does the periodic nature of sine and cosine affect the extrema?
5. Can the method of Lagrange multipliers be applied to constrained extrema?

**Tip:** Always verify if critical points lie within the specified domain when solving optimization problems.

Discuss the function defined as f(x) = sin(x) + 1/2√2 cos(2x) for extreme values in the interval (0, 2π).

Learn how to find the extreme values of the function f(x) = sin(x) + (1/2√2) cos(2x) by using the first derivative and solving for critical points. This problem applies concepts of trigonometry and calculus and is suitable for students in Grades 11-12.

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