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 \( x \)
Given the equation \( x + y = 8 \), we can express \( y \) as:
\[ y = 8 - x \]

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

### Step 3: Differentiate \( f(x) \)
Next, we need to find the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(8x - x^2) = 8 - 2x \]

### Step 4: Set the derivative equal to zero
To find the critical points, set the derivative equal to zero:
\[ 8 - 2x = 0 \]

### Step 5: Solve for \( x \)
Now, solve for \( x \):
\[ 2x = 8 \implies x = 4 \]

### Step 6: Find \( y \) using the constraint
Substituting \( x = 4 \) back into the equation for \( y \):
\[ y = 8 - x = 8 - 4 = 4 \]

### Step 7: Calculate the maximum value of \( xy \)
Now, calculate the maximum value of \( xy \):
\[ xy = 4 \times 4 = 16 \]

### Final Answer
The maximum value of \( xy \) subject to \( x + y = 8 \) is:
\[ \boxed{16} \]

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