To find two numbers whose sum is 10 and for which the sum of their squares is a minimum, let's define the two numbers as $x$ and $y$.

Step 1: Express the sum condition

The condition given is that the sum of the two numbers is 10: $x + y = 10$

Step 2: Express the sum of squares

The sum of the squares of the numbers is: $S = x^2 + y^2$ We need to minimize this expression under the constraint $x + y = 10$.

Step 3: Use substitution

From the sum condition $x + y = 10$, we can express $y$ in terms of $x$: $y = 10 - x$

Now substitute this into the sum of squares: $S = x^2 + (10 - x)^2$

Step 4: Simplify the expression for $S$

Expand $(10 - x)^2$: $S = x^2 + (100 - 20x + x^2)$ $S = 2x^2 - 20x + 100$

Step 5: Minimize the function

To minimize $S$, we take the derivative with respect to $x$ and set it equal to zero: $\frac{dS}{dx} = 4x - 20$ Set the derivative equal to zero to find the critical point: $4x - 20 = 0$ $x = 5$

Step 6: Solve for $y$

Now that we know $x = 5$, substitute this back into the equation $x + y = 10$ to find $y$: $5 + y = 10$ $y = 5$

Step 7: Verify the minimum

The second derivative of $S$ is: $\frac{d^2S}{dx^2} = 4$ Since the second derivative is positive, the function has a minimum at $x = 5$.

Conclusion:

The two numbers are $x = 5$ and $y = 5$, and the sum of their squares is minimized when both numbers are equal. The minimum sum of squares is: $S = 5^2 + 5^2 = 25 + 25 = 50$

Would you like further explanation or have any questions? Here are some related questions:

1. What is the general method to minimize a quadratic function?
2. How does symmetry help solve this problem?
3. What happens to the sum of squares if the sum of the numbers is a different constant (e.g., 20)?
4. Can we apply the same method to maximize the sum of squares?
5. How can we interpret the result geometrically (on a graph)?

Tip: When minimizing or maximizing functions under a constraint, always check if the critical point corresponds to a minimum or maximum using the second derivative test.