If z is a complex number lying in the first quadrant such that `"Re"(z)+"Im"(z)=3`, then the maximum value of `{"Re"(z)}^(2)"Im"(z)`, is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the maximum value of \( \text{Re}(z)^2 \cdot \text{Im}(z) \) given that \( \text{Re}(z) + \text{Im}(z) = 3 \) and \( z \) lies in the first quadrant.

### Step-by-step Solution:

1. **Define the variables**:
Let \( z = x + iy \), where \( x = \text{Re}(z) \) and \( y = \text{Im}(z) \). According to the problem, we have:
\[ x + y = 3 \]

2. **Express \( y \) in terms of \( x \)**:
From the equation \( x + y = 3 \), we can express \( y \) as:
\[ y = 3 - x \]

3. **Substitute \( y \) into the expression**:
We need to maximize the expression \( x^2 \cdot y \). Substituting \( y \) gives us:
\[ x^2 \cdot y = x^2 \cdot (3 - x) = 3x^2 - x^3 \]

4. **Define the function**:
Let \( f(x) = 3x^2 - x^3 \). We will find the maximum value of this function.

5. **Differentiate the function**:
To find the critical points, we differentiate \( f(x) \):
\[ f'(x) = 6x - 3x^2 \]

6. **Set the derivative to zero**:
Setting \( f'(x) = 0 \):
\[ 6x - 3x^2 = 0 \]
Factoring out \( 3x \):
\[ 3x(2 - x) = 0 \]
This gives us the critical points:
\[ x = 0 \quad \text{or} \quad x = 2 \]

7. **Determine the nature of the critical points**:
We will check the second derivative to determine if these points are maxima or minima:
\[ f''(x) = 6 - 6x \]
- At \( x = 0 \):
\[ f''(0) = 6 - 0 = 6 \quad (\text{local minimum}) \]
- At \( x = 2 \):
\[ f''(2) = 6 - 12 = -6 \quad (\text{local maximum}) \]

8. **Calculate the maximum value**:
Now, we calculate \( f(2) \):
\[ f(2) = 3(2^2) - (2^3) = 3 \cdot 4 - 8 = 12 - 8 = 4 \]

9. **Conclusion**:
The maximum value of \( \text{Re}(z)^2 \cdot \text{Im}(z) \) is:
\[ \boxed{4} \]