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We are given the function \( f(x) = \frac{x^2 + A}{x} \), where \( A \) is a constant. We also know that \( f''(2) = \frac{1}{8} \), and we need to find the value of \( A \).

### Step 1: Differentiate \( f(x) \)

Let's start by simplifying the given function:
\[
f(x) = \frac{x^2 + A}{x} = x + \frac{A}{x}
\]

Now, we will differentiate \( f(x) \) to find \( f'(x) \) and then \( f''(x) \).

#### First derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx}\left(x + \frac{A}{x}\right)
\]
Using the power rule:
\[
f'(x) = 1 - \frac{A}{x^2}
\]

#### Second derivative \( f''(x) \):
Now, differentiate \( f'(x) \):
\[
f''(x) = \frac{d}{dx}\left(1 - \frac{A}{x^2}\right)
\]
\[
f''(x) = 0 + \frac{2A}{x^3} = \frac{2A}{x^3}
\]

### Step 2: Use the given condition \( f''(2) = \frac{1}{8} \)

We are told that \( f''(2) = \frac{1}{8} \), so substitute \( x = 2 \) into the expression for \( f''(x) \):
\[
f''(2) = \frac{2A}{2^3} = \frac{2A}{8} = \frac{A}{4}
\]
Set this equal to the given value \( \frac{1}{8} \):
\[
\frac{A}{4} = \frac{1}{8}
\]

### Step 3: Solve for \( A \)

To solve for \( A \), multiply both sides by 4:
\[
A = \frac{4}{8} = \frac{1}{2}
\]

Thus, the value of \( A \) is \( \frac{1}{2} \).

### Final Answer:
\[
A = \frac{1}{2}
\]

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

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Here are 5 related questions to explore:

1. How would you differentiate a function involving higher powers of \( x \) and constants, like \( \frac{x^3 + B}{x} \)?
2. Can you solve a similar problem where the given second derivative is \( \frac{1}{4} \)?
3. How do we apply the chain rule when differentiating more complex functions?
4. What would happen if \( A \) were a function of \( x \) instead of a constant?
5. How would the solution change if you were asked to find \( f'(3) \) instead?

**Tip:** When solving calculus problems, always simplify functions as much as possible before differentiating—it makes the process quicker and reduces errors.

Suppose that f(x) = (x^2 + A) / x, where A is a constant. If f''(2) = 1/8, find the value of A.

Learn how to find the constant A for the function f(x) = (x^2 + A) / x by using second derivative rules. The problem explores key calculus concepts, including differentiation and algebraic manipulation, suitable for advanced high school students.

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