Let's solve this problem step-by-step. Let the two nonnegative numbers be $x$ and $y$.

Step 1: Set up the equations

From the problem statement, we have two conditions:

1. $x + y^2 = 10$
2. We want $x + y$ to be as large as possible.

Step 2: Express $x$ in terms of $y$

From the first equation, we can solve for $x$ in terms of $y$: $x = 10 - y^2$

Step 3: Substitute $x$ in the objective function

We want to maximize $x + y$. Substituting the expression for $x$ from above: $x + y = (10 - y^2) + y$ which simplifies to: $f(y) = 10 + y - y^2$

Step 4: Differentiate $f(y)$ and find the critical points

To maximize $f(y)$, we take the derivative with respect to $y$: $f'(y) = 1 - 2y$ Set the derivative equal to zero to find the critical points: $1 - 2y = 0$ $y = \frac{1}{2}$

Step 5: Find the corresponding value of $x$

Substitute $y = \frac{1}{2}$ back into the equation $x = 10 - y^2$ to find $x$: $x = 10 - \left( \frac{1}{2} \right)^2$ $x = 10 - \frac{1}{4}$ $x = \frac{40}{4} - \frac{1}{4} = \frac{39}{4}$

Step 6: Verify the answer

The numbers that satisfy both conditions are: $x = \frac{39}{4} \quad \text{and} \quad y = \frac{1}{2}$

Conclusion

The correct answer is: (d) $\frac{1}{2}$ and $\frac{39}{4}$

Would you like a more detailed breakdown, or do you have any questions?

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Here are some related questions that expand on this problem:

1. How would the solution change if we wanted to minimize $x + y$ instead of maximizing it?
2. What would happen if the condition $x + y^2 = 10$ were changed to $x \cdot y^2 = 10$?
3. How do we know that our critical point indeed gives a maximum for $x + y$?
4. Can you generalize this method to find the maximum of other expressions with similar constraints?
5. What if we included a third variable in the equation? How would we approach solving for the maximum sum?

Tip: When maximizing or minimizing expressions under constraints, substituting one variable in terms of the other using the constraint can simplify the optimization process.