If A = 4x + `(1)/(x)`, then the value of A + `(1)/(A)` is

Options

A. `(x)/(4x^(2) + 1)`
B. `(4x^(2) + 1)/(x)`
C. `(1)/(4x^(3)+x)`
D. None of these

Answer

To solve the problem, we need to find the value of \( A + \frac{1}{A} \) given that \( A = 4x + \frac{1}{x} \).

### Step-by-Step Solution:

1. **Define the expression for A**:
\[ A = 4x + \frac{1}{x} \]

2. **Find \( \frac{1}{A} \)**:
To find \( \frac{1}{A} \), we first express \( A \) in a single fraction:
\[ A = \frac{4x^2 + 1}{x} \]
Therefore,
\[ \frac{1}{A} = \frac{x}{4x^2 + 1} \]

3. **Add \( A \) and \( \frac{1}{A} \)**:
Now we can add \( A \) and \( \frac{1}{A} \):
\[ A + \frac{1}{A} = \left(4x + \frac{1}{x}\right) + \left(\frac{x}{4x^2 + 1}\right) \]
To combine these, we need a common denominator:
\[ A + \frac{1}{A} = \frac{(4x)(4x^2 + 1) + x}{x(4x^2 + 1)} \]

4. **Simplify the numerator**:
Expanding the numerator:
\[ (4x)(4x^2 + 1) + x = 16x^3 + 4x + x = 16x^3 + 5x \]
So we have:
\[ A + \frac{1}{A} = \frac{16x^3 + 5x}{x(4x^2 + 1)} \]

5. **Final simplification**:
We can simplify the expression:
\[ A + \frac{1}{A} = \frac{16x^3 + 5x}{4x^3 + x} \]

### Final Answer:
Thus, the value of \( A + \frac{1}{A} \) is:
\[ A + \frac{1}{A} = \frac{16x^3 + 5x}{4x^3 + x} \]

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