Let's solve this optimization problem:

Problem Statement

Given non-negative numbers $x$ and $y$ such that $x + y = 9$, we need to:

1. Find the values of $x$ and $y$ that maximize $xy^2$.
2. Determine the maximum value of $xy^2$.

Solution

1. Set Up the Expression to Maximize: We are asked to maximize $f(x, y) = xy^2$ with the constraint $x + y = 9$.

2. Express $x$ in Terms of $y$: From the constraint $x + y = 9$, we can rewrite $x$ as: $x = 9 - y$

   Substituting $x = 9 - y$ into $f(x, y) = xy^2$, we get: $f(y) = (9 - y)y^2 = 9y^2 - y^3$

3. Differentiate with Respect to $y$: To find the maximum, we take the derivative of $f(y)$ with respect to $y$: $f'(y) = \frac{d}{dy}(9y^2 - y^3) = 18y - 3y^2$

   Set $f'(y) = 0$ to find critical points: $18y - 3y^2 = 0$ $3y(6 - y) = 0$ This gives $y = 0$ or $y = 6$.

4. Evaluate $x$ and $f(x, y)$ at Critical Points:

   • When $y = 0$: $x = 9 - 0 = 9$ $f(9, 0) = 9 \cdot 0^2 = 0$

   • When $y = 6$: $x = 9 - 6 = 3$ $f(3, 6) = 3 \cdot 6^2 = 3 \cdot 36 = 108$

5. Conclusion: The maximum value of $xy^2$ is $108$, which occurs when $x = 3$ and $y = 6$.

Answers

1. The values of $x$ and $y$ that maximize $xy^2$ are $x = 3$ and $y = 6$.
2. The maximum value of $xy^2$ is $108$.

Would you like a more detailed explanation of each step, or do you have any specific questions?

Here are some additional questions related to this topic:

1. How would the solution change if the constraint were $x + y = 12$ instead of $x + y = 9$?
2. Can we use the second derivative test to confirm that $y = 6$ gives a maximum?
3. What would happen if we were asked to maximize $xy$ instead of $xy^2$?
4. Could we solve this problem using Lagrange multipliers?
5. How would the solution differ if $x$ and $y$ were allowed to be negative?

Tip: When solving optimization problems with constraints, substituting one variable in terms of another (using the constraint) often simplifies the calculations.