Consider two non-negative numbers  
x
  and  
y
  such that  
x
+
y
=
10
 . What is the maximal value of the quantity  
x
2
y
2
 ?

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.

Explore how to maximize x^2 y^2 given x + y = 10 using calculus and the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Detailed step-by-step solution provided. Suitable for advanced high school students.

Math Problem Statement

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

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