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To solve this problem, let's break down the information given:

We have two positive numbers \( x \) and \( y \) such that:
1. The product of the numbers is 9, so \( x \cdot y = 9 \).
2. We want to find the maximum possible value of \( x + y \).

To find this maximum, we can express \( y \) in terms of \( x \) using the constraint:
\[
y = \frac{9}{x}
\]

Now, we want to maximize \( x + y = x + \frac{9}{x} \).

To find the maximum, let's differentiate \( f(x) = x + \frac{9}{x} \) with respect to \( x \) and set the derivative to zero.

1. Differentiate:
   \[
   f'(x) = 1 - \frac{9}{x^2}
   \]

2. Set \( f'(x) = 0 \):
   \[
   1 - \frac{9}{x^2} = 0
   \]
   \[
   \frac{9}{x^2} = 1
   \]
   \[
   x^2 = 9
   \]
   \[
   x = 3 \quad (\text{since } x \text{ is positive})
   \]

3. Substitute \( x = 3 \) back to find \( y \):
   \[
   y = \frac{9}{3} = 3
   \]

So, \( x = 3 \) and \( y = 3 \) maximize \( x + y \), and their sum is:
\[
x + y = 3 + 3 = 6
\]

### Answer
The largest possible value of \( x + y \) is \( \boxed{6} \).

Would you like further explanation, or have any additional questions?

Here are some related questions:
1. How can we find the minimum possible value of \( x + y \) if \( x \cdot y = 9 \)?
2. How do we determine whether a critical point gives a maximum or minimum?
3. Can the AM-GM inequality be used to solve this type of problem?
4. What if the product constraint was a different number, like 16?
5. How does changing one of the numbers while keeping the product constant affect their sum?

**Tip:** Problems involving maximizing or minimizing sums under a product constraint can often be tackled by using calculus or the AM-GM inequality.

The product of two positive numbers is 9. Find the largest possible value of x + y.

Learn how to find the maximum sum of two positive numbers whose product is 9. This calculus problem uses optimization techniques, including finding the derivative and setting it to zero. Ideal for Grade 11-12 students studying optimization.

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