If `x + y=4` and `x >=0, y>= 0` find the maximum value of `x^3y`.

Options

A. 12
B. 27
C. 20
D. none of these

Answer

To find the maximum value of \( x^3y \) given the constraint \( x + y = 4 \) and \( x \geq 0, y \geq 0 \), we can use the method of Lagrange multipliers or apply the AM-GM inequality. Here, we will use the AM-GM inequality for simplicity.

### Step-by-Step Solution:

1. **Set up the problem**: We have the constraint \( x + y = 4 \). We want to maximize \( z = x^3y \).

2. **Express \( y \) in terms of \( x \)**: From the constraint \( x + y = 4 \), we can express \( y \) as:
\[ y = 4 - x \]

3. **Substitute \( y \) into \( z \)**: Now we substitute \( y \) into the expression for \( z \):
\[ z = x^3(4 - x) = 4x^3 - x^4 \]

4. **Differentiate \( z \) with respect to \( x \)**: To find the maximum, we differentiate \( z \):
\[ \frac{dz}{dx} = 12x^2 - 4x^3 \]

5. **Set the derivative to zero**: To find the critical points, we set the derivative equal to zero:
\[ 12x^2 - 4x^3 = 0 \]
Factoring out \( 4x^2 \):
\[ 4x^2(3 - x) = 0 \]

6. **Solve for \( x \)**: This gives us:
\[ 4x^2 = 0 \quad \Rightarrow \quad x = 0 \]
\[ 3 - x = 0 \quad \Rightarrow \quad x = 3 \]

7. **Find corresponding \( y \) values**: For \( x = 0 \):
\[ y = 4 - 0 = 4 \quad \Rightarrow \quad z = 0^3 \cdot 4 = 0 \]
For \( x = 3 \):
\[ y = 4 - 3 = 1 \quad \Rightarrow \quad z = 3^3 \cdot 1 = 27 \]

8. **Check the endpoints**: Since \( x \) must be non-negative and \( x + y = 4 \), we also check:
- If \( x = 4 \), then \( y = 0 \):
\[ z = 4^3 \cdot 0 = 0 \]

9. **Conclusion**: The maximum value of \( z = x^3y \) occurs at \( x = 3 \) and \( y = 1 \):
\[ \text{Maximum value of } x^3y = 27 \]