The maximum value of `xy` subject to `x +y=8` is :

Options

A. 8
B. 16
C. 20
D. 24

Explanation

To find the maximum value of \( xy \) subject to the constraint \( x + y = 8 \), we can follow these steps:

### Step 1: Express \( y \) in terms of \( x \)
Since we have the equation \( x + y = 8 \), we can express \( y \) as:
\[ y = 8 - x \]

### Step 2: Substitute \( y \) into \( xy \)
Now, we substitute \( y \) into the expression \( xy \):
\[ xy = x(8 - x) = 8x - x^2 \]

### Step 3: Differentiate the expression
Next, we differentiate the expression \( 8x - x^2 \) with respect to \( x \):
\[ \frac{d}{dx}(8x - x^2) = 8 - 2x \]

### Step 4: Set the derivative to zero
To find the maximum value, we set the derivative equal to zero:
\[ 8 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 8 \implies x = 4 \]

### Step 5: Find the corresponding value of \( y \)
Now, we can find the corresponding value of \( y \):
\[ y = 8 - x = 8 - 4 = 4 \]

### Step 6: Calculate the maximum value of \( xy \)
Now, we substitute \( x \) and \( y \) back into the expression for \( xy \):
\[ xy = 4 \cdot 4 = 16 \]

### Conclusion
Thus, the maximum value of \( xy \) subject to the constraint \( x + y = 8 \) is:
\[ \boxed{16} \]