Let's solve the problem step-by-step based on your image.

Part (a): Constructing the Table

We need to find two numbers whose sum is 25 and whose product is a maximum.
For the table, the second number will always be $25 - \text{First number}$, since their sum must always be 25.
The product will then be:

$\text{Product} = (\text{First number}) \times (25 - \text{First number})$

Let's fill in the missing entries in the table.

From the table, the largest product seems to occur when the two numbers are 12 and 13, with a product of 156.

Part (b): Using Calculus to Maximize the Product

Let $x$ be the first number, so the second number is $25 - x$. The product function $P(x)$ is:

$P(x) = x(25 - x) = 25x - x^2$

Now, we find the value of $x$ that maximizes the product by taking the derivative of $P(x)$ and setting it equal to 0.

$P'(x) = 25 - 2x$

Setting $P'(x) = 0$:

$25 - 2x = 0 \quad \Rightarrow \quad 2x = 25 \quad \Rightarrow \quad x = 12.5$

So, the first number is 12.5, and the second number is:

$25 - 12.5 = 12.5$

The maximum product is:

$P(12.5) = 12.5 \times 12.5 = 156.25$

Comparison with Part (a)

In part (a), the closest answer was 12 and 13, giving a product of 156. However, the actual maximum using calculus occurs when both numbers are 12.5, giving a product of 156.25.

Final Answers

1. Estimated answer from part (a): 12, 13
2. Answer from part (b) using calculus: 12.5, 12.5

Do you have any questions or want further details on this?

Here are five related questions to deepen your understanding:

1. What is the role of the derivative in maximizing functions?
2. Why does the product increase as the numbers become closer to each other?
3. What happens if the sum constraint changes? For example, sum = 30.
4. How would the problem change if the numbers must be integers?
5. What does the second derivative tell us about the nature of the maximum?

Tip: Always use derivatives to find precise maxima or minima in optimization problems!