If `f(x)=root (3)(8x^(3)+mx^(2))-nx` such that `lim_(xrarroo)f(x)=1` then (A) `m+n=15` (B) `m-n=10` (C) `m-n=12` (D) `m+n=14`

Options

A. `m+n=15`
B. `m-n=10`
C. `m-n=12`
D. `m+n=14`

Explanation

To solve the problem, we need to analyze the function \( f(x) = \sqrt[3]{8x^3 + mx^2} - nx \) and find the values of \( m \) and \( n \) such that \( \lim_{x \to \infty} f(x) = 1 \).

### Step-by-Step Solution:

1. **Rewrite the Function**:
\[ f(x) = \sqrt[3]{8x^3 + mx^2} - nx \]

2. **Find the Limit as \( x \to \infty \)**:
To analyze the limit, we can factor out \( x^3 \) from the cube root:
\[ f(x) = \sqrt[3]{x^3(8 + \frac{m}{x})} - nx = x \left( \sqrt[3]{8 + \frac{m}{x}} - n \right) \]

3. **Evaluate the Limit**:
As \( x \to \infty \), \( \frac{m}{x} \to 0 \), so:
\[ \sqrt[3]{8 + \frac{m}{x}} \to \sqrt[3]{8} = 2 \]
Therefore,
\[ f(x) \approx x(2 - n) \]

4. **Set the Limit Equal to 1**:
For the limit to equal 1, we need:
\[ \lim_{x \to \infty} x(2 - n) = 1 \]
This implies that \( 2 - n = 0 \) (otherwise the limit would go to infinity), hence:
\[ n = 2 \]

5. **Substitute \( n \) Back into the Limit**:
Now substituting \( n = 2 \) back into the limit expression:
\[ f(x) = \sqrt[3]{8x^3 + mx^2} - 2x \]
We rewrite it as:
\[ f(x) = x \left( \sqrt[3]{8 + \frac{m}{x}} - 2 \right) \]

6. **Finding \( m \)**:
We need to ensure that:
\[ \lim_{x \to \infty} x \left( \sqrt[3]{8 + \frac{m}{x}} - 2 \right) = 1 \]
Using the expansion \( \sqrt[3]{8 + \frac{m}{x}} \approx 2 + \frac{m}{3 \cdot 2^2 x} \) for large \( x \):
\[ \sqrt[3]{8 + \frac{m}{x}} - 2 \approx \frac{m}{12x} \]
Thus,
\[ x \cdot \frac{m}{12x} = \frac{m}{12} \]
Setting this equal to 1 gives:
\[ \frac{m}{12} = 1 \implies m = 12 \]

7. **Calculate \( m + n \) and \( m - n \)**:
Now we have \( m = 12 \) and \( n = 2 \):
\[ m + n = 12 + 2 = 14 \]
\[ m - n = 12 - 2 = 10 \]

### Final Results:
- \( m + n = 14 \)
- \( m - n = 10 \)

### Conclusion:
The correct options are:
- (A) \( m+n=15 \) - Incorrect
- (B) \( m-n=10 \) - Correct
- (C) \( m-n=12 \) - Incorrect
- (D) \( m+n=14 \) - Correct