If `x+ y= 6` then max value of xy

Answer

To find the maximum value of \( xy \) given that \( x + y = 6 \), we can use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Here’s the step-by-step solution:

### Step 1: Understand the relationship
We know that:
\[ x + y = 6 \]
We want to maximize the product:
\[ xy \]

### Step 2: Apply the AM-GM inequality
According to the AM-GM inequality, the arithmetic mean of two non-negative numbers is greater than or equal to their geometric mean. Therefore, we can write:
\[ \frac{x + y}{2} \geq \sqrt{xy} \]

### Step 3: Substitute the known value
Substituting \( x + y = 6 \) into the inequality gives:
\[ \frac{6}{2} \geq \sqrt{xy} \]
This simplifies to:
\[ 3 \geq \sqrt{xy} \]

### Step 4: Square both sides
To eliminate the square root, we square both sides of the inequality:
\[ 3^2 \geq xy \]
This results in:
\[ 9 \geq xy \]

### Step 5: Conclusion
The maximum value of \( xy \) is \( 9 \). Therefore, the maximum value of the product \( xy \) when \( x + y = 6 \) is:
\[ \boxed{9} \]