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To find the absolute extrema of the function \( y = 7 \cos(x) \) on the closed interval \( [0, 2\pi] \), we'll go through the following steps:

### Step 1: Identify the function
The function is given as:
\[
y = 7 \cos(x)
\]

### Step 2: Find the derivative of \( y \)
To locate the critical points, we need to take the derivative of \( y \) with respect to \( x \):
\[
y' = -7 \sin(x)
\]

### Step 3: Set the derivative to zero to find critical points
We set \( y' = 0 \) to find the critical points within the interval \( [0, 2\pi] \):
\[
-7 \sin(x) = 0
\]
\[
\sin(x) = 0
\]
The values of \( x \) where \( \sin(x) = 0 \) in the interval \( [0, 2\pi] \) are:
\[
x = 0, \pi, \text{ and } 2\pi
\]

### Step 4: Evaluate \( y \) at the critical points and endpoints
Now, we evaluate \( y = 7 \cos(x) \) at \( x = 0 \), \( x = \pi \), and \( x = 2\pi \).

1. **At \( x = 0 \):**
   \[
   y = 7 \cos(0) = 7 \times 1 = 7
   \]

2. **At \( x = \pi \):**
   \[
   y = 7 \cos(\pi) = 7 \times (-1) = -7
   \]

3. **At \( x = 2\pi \):**
   \[
   y = 7 \cos(2\pi) = 7 \times 1 = 7
   \]

### Step 5: Determine the absolute extrema
- The maximum value of \( y \) is 7, which occurs at both \( x = 0 \) and \( x = 2\pi \).
- The minimum value of \( y \) is -7, which occurs at \( x = \pi \).

### Final Answer
- **Minimum:** \( (x, y) = (\pi, -7) \)
- **Maximum (smaller \( x \)-value):** \( (x, y) = (0, 7) \)
- **Maximum (larger \( x \)-value):** \( (x, y) = (2\pi, 7) \)

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to expand your understanding:

1. How do we determine critical points for functions involving trigonometric functions?
2. What are the key differences between absolute and relative extrema?
3. Why does \( \cos(x) \) have a range of \( [-1, 1] \), and how does it affect finding extrema?
4. How would the extrema change if the function were \( y = 7 \sin(x) \) instead of \( y = 7 \cos(x) \)?
5. What is the significance of endpoints in finding absolute extrema on a closed interval?

**Tip:** When working with trigonometric functions on specific intervals, sketching the function's graph can provide visual insight into where maxima and minima occur.

Find the absolute extrema of the function y = 7 cos(x) on the closed interval [0, 2π].

Learn how to find the absolute extrema of the trigonometric function y = 7 cos(x) on the closed interval [0, 2π]. This solution covers finding critical points by taking the derivative and evaluating endpoints, demonstrating key calculus concepts for Grades 11-12 or college students.

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