Let a and b be positive real numbers with ` a^(3) +b^(3) = a -b and k = a^(2) + 4b^(2)` , then (1)` k lt 1` (2) `k gt 1` (3) `k = 1 `(4) `k gt 2`

Answer

To solve the problem, we need to analyze the given equations step by step.

### Step 1: Understand the given equations
We have two equations:
1. \( a^3 + b^3 = a - b \)
2. \( k = a^2 + 4b^2 \)

### Step 2: Rewrite the first equation
Using the identity for the sum of cubes, we can rewrite the first equation:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Thus, we can express the equation as:
\[ (a + b)(a^2 - ab + b^2) = a - b \]

### Step 3: Analyze the equation
Since \( a \) and \( b \) are positive real numbers, we can analyze the behavior of the left-hand side and the right-hand side. The left-hand side is always positive, and we need to find conditions under which the equation holds.

### Step 4: Consider the case when \( b = 0 \)
If we set \( b = 0 \), we have:
\[ a^3 = a \]
This implies:
\[ a(a^2 - 1) = 0 \implies a = 1 \quad (\text{since } a > 0) \]
Thus, when \( b = 0 \), \( a = 1 \).

### Step 5: Calculate \( k \)
Substituting \( a = 1 \) and \( b = 0 \) into the equation for \( k \):
\[ k = a^2 + 4b^2 = 1^2 + 4(0)^2 = 1 \]

### Step 6: Consider the case when \( a = 0 \)
If we set \( a = 0 \), we have:
\[ b^3 = -b \]
This is not possible since \( b \) is positive.

### Step 7: Analyze the behavior of \( k \)
Now we need to analyze the behavior of \( k \) when both \( a \) and \( b \) are positive. Since the equation \( a^3 + b^3 = a - b \) must hold, we can infer that as \( b \) increases, \( a \) must decrease to maintain the equality.

### Step 8: Conclusion
From our analysis, we found that when \( b = 0 \), \( k = 1 \). As \( b \) increases, \( k \) must remain less than 1 to satisfy the equation \( a^3 + b^3 = a - b \). Therefore, we conclude:
\[ k < 1 \]

### Final Answer
Thus, the correct option is:
(1) \( k < 1 \)