Consider two non-negative numbers
x and
y such that
x + y

To find the maximal value of the quantity $x^2 y^2$ given that $x + y = 10$, we can use the method of calculus or the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

Step 1: Express the function to maximize

We are given: $x + y = 10$ We want to maximize: $f(x, y) = x^2 y^2$ We can express $y$ in terms of $x$: $y = 10 - x$ Thus, the function becomes: $f(x) = x^2 (10 - x)^2$

Step 2: Simplify the function

Let's expand the function: $f(x) = x^2 (100 - 20x + x^2)$ $f(x) = x^2 (x^2 - 20x + 100)$ $f(x) = x^4 - 20x^3 + 100x^2$

Step 3: Take the derivative and find critical points

To maximize $f(x)$, take the derivative with respect to $x$: $f'(x) = 4x^3 - 60x^2 + 200x$ Set the derivative equal to zero to find critical points: $4x^3 - 60x^2 + 200x = 0$ Factor out $4x$: $4x(x^2 - 15x + 50) = 0$ This gives us: $x = 0 \quad \text{or} \quad x^2 - 15x + 50 = 0$

Step 4: Solve the quadratic equation

For the quadratic equation: $x^2 - 15x + 50 = 0$ Use the quadratic formula: $x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(1)(50)}}{2(1)}$ $x = \frac{15 \pm \sqrt{225 - 200}}{2}$ $x = \frac{15 \pm \sqrt{25}}{2}$ $x = \frac{15 \pm 5}{2}$ Thus: $x = 10 \quad \text{or} \quad x = 5$

Step 5: Evaluate the function at critical points

For $x = 0$: $f(0) = 0^2 \times (10 - 0)^2 = 0$ For $x = 10$: $f(10) = 10^2 \times (10 - 10)^2 = 0$ For $x = 5$ (and $y = 5$): $f(5) = 5^2 \times 5^2 = 25 \times 25 = 625$

Conclusion

The maximal value of $x^2 y^2$ occurs when $x = 5$ and $y = 5$, and the maximal value is: $\boxed{625}$

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. What is the minimal value of $x^2 y^2$ given the same conditions?
2. How would the maximal value change if $x + y = 20$ instead of 10?
3. Can the AM-GM inequality be used directly to solve this problem? How?
4. What happens if $x + y$ is a variable instead of a fixed number?
5. How can this problem be extended to maximize $x^n y^n$ for any positive integer $n$?

Tip: Always check if there is a simpler method, like using inequalities, before jumping into calculus for optimization problems.