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The problem asks to find the two values of \( x \) in the interval \([0, 2\pi]\) at which the graph of \( f(x) = x - 2 \sin x \) has a horizontal tangent.

### Solution Outline:
A horizontal tangent occurs where the derivative of the function \( f(x) \) is equal to 0. So, we will:
1. Differentiate \( f(x) \).
2. Set the derivative equal to 0 and solve for \( x \) within the given interval.

### Step 1: Differentiate \( f(x) \)
We need to compute \( f'(x) \), the derivative of \( f(x) = x - 2 \sin x \).
\[
f'(x) = \frac{d}{dx}(x) - 2 \cdot \frac{d}{dx}(\sin x) = 1 - 2 \cos x
\]

### Step 2: Set the derivative equal to 0
To find the points where the graph has a horizontal tangent, we set \( f'(x) = 0 \):
\[
1 - 2 \cos x = 0
\]
Solve for \( \cos x \):
\[
2 \cos x = 1 \quad \Rightarrow \quad \cos x = \frac{1}{2}
\]

### Step 3: Solve for \( x \)
The general solutions for \( \cos x = \frac{1}{2} \) occur at:
\[
x = \frac{\pi}{3} \quad \text{and} \quad x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}
\]

### Step 4: Check the interval
Both \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \) lie within the interval \([0, 2\pi]\).

### Final Answer:
The two values of \( x \) where the graph of \( f(x) = x - 2 \sin x \) has a horizontal tangent are:
\[
x = \frac{\pi}{3} \quad \text{and} \quad x = \frac{5\pi}{3}
\]

Would you like more details or have any questions?

Here are five related questions:
1. What does it mean for a function's graph to have a horizontal tangent?
2. How do we determine the points where a graph has a horizontal tangent for any given function?
3. What other types of tangents can we find apart from horizontal ones?
4. Can you explain how to solve trigonometric equations like \( \cos x = \frac{1}{2} \)?
5. What role does differentiation play in identifying tangents on a curve?

**Tip**: When differentiating trigonometric functions, remember that the derivative of \( \sin x \) is \( \cos x \), and the derivative of \( \cos x \) is \( -\sin x \).

Find the two values of x in the interval [0, 2π] at which the graph of f(x) = x - 2sin(x) has a horizontal tangent.

Learn how to find the values of x in the interval [0, 2π] where the graph of f(x) = x - 2sin(x) has a horizontal tangent. This solution involves differentiating the function and solving trigonometric equations, suitable for students in Grades 11-12.

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