If x+y =12 what is the maximum value of xy?

Options

A. 25
B. 36
C. 49
D. 64

Explanation

To find the maximum value of \( xy \) given that \( x + y = 12 \), we can follow these steps:

### Step 1: Express \( y \) in terms of \( x \)
From the equation \( x + y = 12 \), we can express \( y \) as:
\[ y = 12 - x \]

### Step 2: Write the function \( f(x) \)
Now, we can write the function \( f(x) \) that we want to maximize:
\[ f(x) = xy = x(12 - x) = 12x - x^2 \]

### Step 3: Find the derivative of \( f(x) \)
Next, we need to find the first derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(12x - x^2) = 12 - 2x \]

### Step 4: Set the derivative to zero to find critical points
To find the maximum value, we set the first derivative equal to zero:
\[ 12 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 12 \implies x = 6 \]

### Step 5: Find the corresponding value of \( y \)
Now that we have \( x = 6 \), we can find \( y \):
\[ y = 12 - x = 12 - 6 = 6 \]

### Step 6: Calculate the maximum value of \( xy \)
Now we can calculate the maximum value of \( xy \):
\[ xy = 6 \times 6 = 36 \]

### Conclusion
Thus, the maximum value of \( xy \) is:
\[ \boxed{36} \]

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