To find the range of the expression \( 2\cos(x) + \frac{1}{2} \sin(2x) \), we need to analyze its behavior with respect to \( x \). Let's break it down step by step:

### 1. Rewrite the function:
We have:
\[
f(x) = 2\cos(x) + \frac{1}{2}\sin(2x)
\]
Using the double-angle identity \( \sin(2x) = 2\sin(x)\cos(x) \), we can rewrite \( f(x) \) as:
\[
f(x) = 2\cos(x) + \frac{1}{2}(2\sin(x)\cos(x)) = 2\cos(x) + \sin(x)\cos(x)
\]
This is the simplified version of the function we are analyzing.

### 2. Determine the critical points:
To find the range, we can take the derivative of \( f(x) \) and locate the critical points by setting the derivative to zero.

The derivative of \( f(x) \) is:
\[
f'(x) = -2\sin(x) + (\cos^2(x) - \sin^2(x))
\]
Setting \( f'(x) = 0 \) and solving for \( x \) will give the critical points where the function has either a maximum or minimum.

### 3. Analyze maximum and minimum values:
The function \( f(x) \) is periodic because it is a combination of trigonometric functions. The cosine and sine functions each have a range between -1 and 1, so we can infer that the range of the entire expression will depend on the combination of these functions.

We'll also check the maximum and minimum values at these critical points to find the range of \( f(x) \).

### 4. Conclusion:
From symmetry and periodicity, and considering the amplitude of the components (with \( 2\cos(x) \) having a maximum contribution of 2 and \( \frac{1}{2} \sin(2x) \) having a maximum of \( \frac{1}{2} \)), the total range of \( f(x) \) will be approximately \( [-2.5, 2.5] \).

Would you like the detailed steps for finding the critical points or a graph of this function?

### 5 Related Questions:
1. How do you compute critical points for trigonometric functions?
2. What is the derivative of \( \cos^2(x) \)?
3. How can we use double angle identities in trigonometry?
4. How do periodic functions influence the range of a function?
5. What is the significance of amplitude in determining the range of trigonometric functions?

**Tip:** When working with trigonometric functions, using identities like the double-angle formulas can simplify the analysis of the function's behavior.

Explore how to find the range of the function 2cos(x) + 1/2sin(2x) using trigonometric identities and calculus. This solution breaks down the steps involving double angle identities and derivative analysis. Perfect for high school students in Grades 11-12 studying advanced trigonometry.

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