Let `a`, `b gt 0` satisfies `a^(3)+b^(3)=a-b`. Then

Options

A. `a^(2)+b^(2)=1`
B. `a^(2)+ab+b^(2) lt 1`
C. `a^(2)+b^(2) gt 1`
D. none of these

Explanation

To solve the problem given that \( a, b > 0 \) and \( a^3 + b^3 = a - b \), we need to find the relation involving \( a^2 + b^2 + ab \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ a^3 + b^3 = a - b \]

2. **Use the identity for the sum of cubes**:
The identity for the sum of cubes is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Therefore, we can rewrite the equation as:
\[ (a + b)(a^2 - ab + b^2) = a - b \]

3. **Rearranging the equation**:
We can express \( a - b \) as:
\[ a - b = (a + b)(a^2 - ab + b^2) \]
This implies:
\[ a^2 - ab + b^2 = \frac{a - b}{a + b} \]

4. **Express \( a^2 + b^2 + ab \)**:
We know:
\[ a^2 + b^2 + ab = (a^2 - ab + b^2) + 2ab \]
Substituting from our previous step:
\[ a^2 + b^2 + ab = \frac{a - b}{a + b} + 2ab \]

5. **Analyze the expression**:
Since \( a, b > 0 \), both \( a + b \) and \( a - b \) are positive or negative depending on the relationship between \( a \) and \( b \). However, since we are given \( a^3 + b^3 = a - b \), we can analyze the bounds of the expression.

6. **Finding bounds**:
We know that \( a^3 + b^3 \) is positive, and since \( a - b \) can be negative or positive, we need to analyze the implications of this. Given that \( a^3 + b^3 \) must equal \( a - b \), we can conclude that:
\[ a^3 + b^3 < 1 \quad \text{(since both terms are positive)} \]
This leads us to conclude that:
\[ a^2 + b^2 + ab < 1 \]

### Conclusion:
Thus, we have shown that:
\[ a^2 + b^2 + ab < 1 \]