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To find the absolute extreme values of the function \( f(x) = \ln(x + 2) + \frac{1}{x} \) over the interval \( 1 \leq x \leq 10 \), we need to:

1. **Identify the critical points** by finding where the derivative \( f'(x) \) is zero or undefined.
2. **Evaluate the function** at those critical points and at the endpoints \( x = 1 \) and \( x = 10 \) to determine the absolute maximum and minimum.

### Step 1: Derivative of \( f(x) \)

Given:
\[
f(x) = \ln(x + 2) + \frac{1}{x}
\]

The derivative \( f'(x) \) is:
\[
f'(x) = \frac{1}{x + 2} - \frac{1}{x^2}
\]

To find critical points, set \( f'(x) = 0 \):
\[
\frac{1}{x + 2} - \frac{1}{x^2} = 0
\]
\[
\frac{1}{x + 2} = \frac{1}{x^2}
\]
Cross-multiplying gives:
\[
x^2 = x + 2
\]
\[
x^2 - x - 2 = 0
\]
\[
(x - 2)(x + 1) = 0
\]

Thus, \( x = 2 \) and \( x = -1 \) are solutions. However, since \( x = -1 \) is outside the interval \( [1, 10] \), we discard it. The only critical point within the interval is \( x = 2 \).

### Step 2: Evaluate \( f(x) \) at \( x = 1 \), \( x = 2 \), and \( x = 10 \)

1. **At \( x = 1 \):**
   \[
   f(1) = \ln(1 + 2) + \frac{1}{1} = \ln(3) + 1
   \]

2. **At \( x = 2 \):**
   \[
   f(2) = \ln(2 + 2) + \frac{1}{2} = \ln(4) + \frac{1}{2}
   \]

3. **At \( x = 10 \):**
   \[
   f(10) = \ln(10 + 2) + \frac{1}{10} = \ln(12) + \frac{1}{10}
   \]

### Step 3: Determine Maximum and Minimum Values

Comparing the values:
- \( f(1) = \ln(3) + 1 \)
- \( f(2) = \ln(4) + \frac{1}{2} \)
- \( f(10) = \ln(12) + \frac{1}{10} \)

Since \( \ln(4) + \frac{1}{2} \) (at \( x = 2 \)) is the smallest value, and \( \ln(12) + \frac{1}{10} \) (at \( x = 10 \)) is the largest value, we conclude:

- **Minimum value**: \( \ln(4) + \frac{1}{2} \) at \( x = 2 \)
- **Maximum value**: \( \ln(12) + \frac{1}{10} \) at \( x = 10 \)

### Answer

The correct option is:
**A. Minimum value is \( \ln 4 + \frac{1}{2} \) at \( x = 2 \); maximum value is \( \ln 12 + \frac{1}{10} \) at \( x = 10 \).**

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Would you like a detailed breakdown of any part of this solution or have other questions?

#### Follow-Up Questions:
1. What is the significance of critical points in finding absolute extrema?
2. How does the natural logarithm function impact the shape of \( f(x) \)?
3. Why do we only consider the interval endpoints and critical points for extreme values?
4. Can absolute extrema occur at points where the derivative does not exist?
5. What other methods could be used to confirm these extreme values?

#### Tip:
When analyzing extrema on a closed interval, always evaluate the function at both critical points and endpoints, as the absolute extrema may occur at any of these locations.

Find the absolute extreme values of the function on the interval. f(x) = ln(x + 2) + 1/x, 1 ≤ x ≤ 10. Select the correct option with the minimum and maximum values from A, B, C, or D.

Learn how to find the absolute maximum and minimum values of the function f(x) = ln(x + 2) + 1/x over the interval [1, 10]. This problem involves calculus concepts such as derivatives, critical points, and the Extreme Value Theorem. Suitable for college-level calculus students.

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