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To solve this problem, we need to find the linearization \( L(x) \) of the function \( f(x) \) at \( x = a \).

### 1. **Function and Point of Linearization**:
We are given:
\[
f(x) = x + \frac{1}{x}, \quad a = 1.
\]
We will linearize \( f(x) \) around \( x = 1 \).

### 2. **Formula for Linearization**:
The linearization of a function \( f(x) \) at \( x = a \) is given by the formula:
\[
L(x) = f(a) + f'(a)(x - a).
\]
So, we need to calculate \( f(a) \) and \( f'(a) \), where \( a = 1 \).

### 3. **Step 1: Calculate \( f(a) \)**:
Substitute \( a = 1 \) into \( f(x) \):
\[
f(1) = 1 + \frac{1}{1} = 2.
\]

### 4. **Step 2: Calculate the Derivative \( f'(x) \)**:
Differentiate \( f(x) \):
\[
f'(x) = \frac{d}{dx}\left(x + \frac{1}{x}\right) = 1 - \frac{1}{x^2}.
\]
Now, evaluate \( f'(x) \) at \( x = 1 \):
\[
f'(1) = 1 - \frac{1}{1^2} = 0.
\]

### 5. **Step 3: Write the Linearization \( L(x) \)**:
Using the linearization formula:
\[
L(x) = f(1) + f'(1)(x - 1).
\]
Substitute \( f(1) = 2 \) and \( f'(1) = 0 \):
\[
L(x) = 2 + 0(x - 1) = 2.
\]

### Final Answer:
The linearization of \( f(x) \) at \( x = 1 \) is:
\[
L(x) = 2.
\]

Would you like further details or clarifications on any part of the solution?

### Related Questions:
1. How do you interpret the result of a linearization geometrically?
2. What happens to the linearization if \( f'(a) \neq 0 \)?
3. Can you linearize the function \( f(x) = x + \frac{1}{x} \) at another point, say \( a = 2 \)?
4. How does the linearization approximate the function near \( x = a \)?
5. How would the linearization change if the function had higher-order terms?

### Tip:
Linearization is essentially finding the equation of the tangent line to the function at a given point. It approximates the function well when you're close to that point!

Find the linearization L(x) of f(x) at x = a, where f(x) = x + 1/x, a = 1.

Learn how to find the linearization of the function f(x) = x + 1/x at the point x = 1. This problem involves key calculus concepts, including differentiation and linear approximation, and is suitable for students in Grades 11-12.

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