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To determine the absolute extreme values of the function \( f(x) = 5 \cos^2(x) \) on the interval \([0, \pi]\), we need to follow these steps:

### Step 1: Find the critical points
The critical points are values of \( x \) where \( f'(x) = 0 \) or where \( f'(x) \) is undefined.

1. **Rewrite \( f(x) \):**
   \[
   f(x) = 5 \cos^2(x)
   \]
   
2. **Differentiate \( f(x) \):**
   Using the chain rule, we differentiate \( f(x) \) with respect to \( x \):
   \[
   f'(x) = 5 \cdot 2 \cos(x) \cdot (-\sin(x)) = -10 \cos(x) \sin(x)
   \]
   
   Using the double-angle identity, \( \sin(2x) = 2 \sin(x) \cos(x) \), we can simplify this as:
   \[
   f'(x) = -5 \sin(2x)
   \]
   
3. **Set \( f'(x) = 0 \) and solve for \( x \):**
   \[
   -5 \sin(2x) = 0
   \]
   Dividing by \(-5\), we get:
   \[
   \sin(2x) = 0
   \]
   
   The solutions for \( \sin(2x) = 0 \) on \([0, \pi]\) are:
   \[
   2x = 0, \pi, 2\pi
   \]
   which gives:
   \[
   x = 0, \frac{\pi}{2}, \pi
   \]

### Step 2: Evaluate \( f(x) \) at the critical points and endpoints
Now we calculate \( f(x) \) at \( x = 0 \), \( x = \frac{\pi}{2} \), and \( x = \pi \) to determine the absolute minimum and maximum values.

1. **At \( x = 0 \):**
   \[
   f(0) = 5 \cos^2(0) = 5 \cdot 1 = 5
   \]

2. **At \( x = \frac{\pi}{2} \):**
   \[
   f\left(\frac{\pi}{2}\right) = 5 \cos^2\left(\frac{\pi}{2}\right) = 5 \cdot 0 = 0
   \]

3. **At \( x = \pi \):**
   \[
   f(\pi) = 5 \cos^2(\pi) = 5 \cdot 1 = 5
   \]

### Step 3: Determine the absolute extreme values
From the evaluations above:
- The absolute maximum value of \( f(x) \) on \([0, \pi]\) is \( 5 \), occurring at \( x = 0 \) and \( x = \pi \).
- The absolute minimum value of \( f(x) \) on \([0, \pi]\) is \( 0 \), occurring at \( x = \frac{\pi}{2} \).

### Final Answer
- **Absolute Maximum**: \( f(x) = 5 \) at \( x = 0 \) and \( x = \pi \).
- **Absolute Minimum**: \( f(x) = 0 \) at \( x = \frac{\pi}{2} \).

Would you like more details on any step, or do you have any questions?

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Here are some related questions to consider:
1. How do we use trigonometric identities to simplify derivatives?
2. Why is it necessary to evaluate endpoints when finding absolute extrema on a closed interval?
3. How would the answer change if the interval were \((0, \pi)\) instead of \([0, \pi]\)?
4. What are the double-angle identities, and why are they useful in trigonometry?
5. How can we verify the nature (maximum or minimum) of a critical point?

**Tip:** Always consider the interval boundaries when looking for absolute extrema on a closed interval; they can sometimes yield extreme values not found by only checking critical points.

Determine the location and value of the absolute extreme values of f on the given interval, if they exist. f(x) = 5 cos^2(x) on [0, π]

Learn how to find the absolute extrema of f(x) = 5 cos^2(x) on the interval [0, π]. This problem demonstrates critical point analysis, derivative calculation using trigonometric identities, and evaluation of endpoints. Suitable for college calculus students.

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