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To find the global maximum and minimum of the function \( f(x) = x^{10} - 20x \) over the interval \( 0 \leq x \leq 3.2 \), we need to analyze its behavior through calculus. Here’s the approach to solving it:

### Step 1: Find the derivative of \( f(x) \)
To locate critical points, we first differentiate \( f(x) \) with respect to \( x \):
\[
f'(x) = 10x^9 - 20
\]

### Step 2: Set the derivative to zero
To find the critical points, we set \( f'(x) = 0 \):
\[
10x^9 - 20 = 0
\]
\[
x^9 = 2
\]
\[
x = 2^{1/9} \approx 1.08
\]

### Step 3: Evaluate \( f(x) \) at critical points and endpoints
We evaluate \( f(x) \) at \( x = 0 \), \( x = 1.08 \) (the critical point), and \( x = 3.2 \) (the upper boundary of the interval).

1. **At \( x = 0 \):**
   \[
   f(0) = 0^{10} - 20 \cdot 0 = 0
   \]

2. **At \( x = 1.08 \):**
   \[
   f(1.08) = (1.08)^{10} - 20 \cdot 1.08 \approx -17.38
   \]

3. **At \( x = 3.2 \):**
   \[
   f(3.2) = (3.2)^{10} - 20 \cdot 3.2 \approx 984.92
   \]

### Step 4: Determine global maximum and minimum
- The **global maximum** occurs at \( x = 3.2 \), with \( f(3.2) \approx 984.92 \).
- The **global minimum** occurs at \( x = 1.08 \), with \( f(1.08) \approx -17.38 \).

### Answer:
- **Global Maximum**: \( (x, y) = (3.2, 984.92) \)
- **Global Minimum**: \( (x, y) = (1.08, -17.38) \)

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

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Here are some related questions to extend your understanding:

1. What are the conditions for identifying a global maximum and minimum in a closed interval?
2. Why is it important to check both endpoints and critical points in a closed interval problem?
3. How would the solution change if the interval were open, for example, \( 0 < x < 3.2 \)?
4. How do we verify that a critical point is indeed a minimum or maximum?
5. What role does the second derivative play in determining the nature of critical points?

**Tip:** Always verify that critical points lie within the specified interval, as values outside this range are not relevant for the solution.

Find the points at which y = f(x) = x^10 - 20x has a global maximum and minimum on the interval 0 ≤ x ≤ 3.2. Round your answers to two decimal places.

Find the global maximum and minimum of the function f(x) = x^10 - 20x on the interval 0 ≤ x ≤ 3.2. This calculus problem demonstrates how to use derivatives and critical points to determine extrema. Suitable for students in Grades 11-12.

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