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 where \( A = 4x + \frac{1}{x} \) and we need to find the value of \( A + \frac{1}{A} \), we can follow these steps:

### Step 1: Write down the expression for \( A \)
Given:
\[ A = 4x + \frac{1}{x} \]

### Step 2: Find \( \frac{1}{A} \)
To find \( \frac{1}{A} \), we will take the reciprocal of \( A \):
\[ \frac{1}{A} = \frac{1}{4x + \frac{1}{x}} \]
To simplify this, we can multiply the numerator and the denominator by \( x \):
\[ \frac{1}{A} = \frac{x}{4x^2 + 1} \]

### Step 3: Find \( A + \frac{1}{A} \)
Now we can find \( A + \frac{1}{A} \):
\[ A + \frac{1}{A} = \left(4x + \frac{1}{x}\right) + \frac{x}{4x^2 + 1} \]
To combine these fractions, we need a common denominator. The common denominator will be \( x(4x^2 + 1) \):
\[ A + \frac{1}{A} = \frac{(4x + \frac{1}{x}) \cdot (4x^2 + 1) + x}{x(4x^2 + 1)} \]

### Step 4: Simplify the numerator
Expanding the numerator:
\[ = \frac{(4x)(4x^2 + 1) + \left(\frac{1}{x}\right)(4x^2 + 1) + x}{x(4x^2 + 1)} \]
\[ = \frac{16x^3 + 4x + \frac{4}{x} + 1 + x}{x(4x^2 + 1)} \]
Combining like terms:
\[ = \frac{16x^3 + 5x + \frac{4}{x} + 1}{x(4x^2 + 1)} \]

### Step 5: Final expression
Now we can write the final expression:
\[ A + \frac{1}{A} = \frac{16x^3 + 5x + 1 + \frac{4}{x}}{x(4x^2 + 1)} \]

### Step 6: Simplify further if possible
To simplify further, we can multiply through by \( x \) to eliminate the fraction:
\[ = \frac{16x^4 + 5x^2 + 4 + 1}{4x^3 + x} \]
\[ = \frac{16x^4 + 5x^2 + 5}{4x^3 + x} \]

### Conclusion
Thus, the value of \( A + \frac{1}{A} \) is:
\[ A + \frac{1}{A} = \frac{16x^4 + 5x^2 + 5}{4x^3 + x} \]