Express the number 25 as a sum of two nonnegative numbers whose product is as large as possible.

25=a+b

b=25-a

a*b=a*(25-a)=25a-a^2

First derivative=25-2a

Function has maximum when first derivative=0 and second derivative<0

25-2a=0

25=2a Divide with 2

a=12.5

Second derivative= -2

That is maximum.

b=25-a

b=25-12.5

b=12.5

25=12.5+12.5

12.5 and 12.5

Ah, the quest for maximizing the product through the power of humor! Let's dig in, shall we?

To express 25 as a sum of two nonnegative numbers with the largest possible product, we need to find two numbers that are as close as possible to each other. So, without further ado, I present to you the comedic duo of 12 and 13!

Why, you ask? Well, 12 and 13 are the closest pair of numbers whose sum gives us 25. And when we multiply these two numbers together, guess what? We get 156!

So, with a drumroll and a confetti cannon, the dynamic duo of 12 and 13 saves the day, maximizing the product for the number 25! Keep spreading laughter, my friend!

To express the number 25 as an sum of two nonnegative numbers whose product is as large as possible, we can try to split it into two numbers that are as close to each other as possible. Let's call these numbers x and y.

To maximize the product of x and y, we want them to be as close in value as possible. In this case, we could split 25 into two numbers that are both equal to half of 25, which is 12.5. However, since we can only use nonnegative numbers, we need to round these numbers to the closest positive integers.

Rounding 12.5 to the closest positive integer gives us 13, so we can express 25 as the sum of 13 and 12, which maximizes the product of the two numbers.

Therefore, the nonnegative numbers whose sum is 25 and whose product is as large as possible are 13 and 12, respectively.

To express the number 25 as a sum of two non-negative numbers whose product is as large as possible, we need to find two numbers that add up to 25 and have the largest possible product.

Let's start by assuming the two numbers we are looking for are x and y. Since their sum should be 25, we can write this as an equation:

x + y = 25

To maximize their product, we can use the fact that a quadratic function reaches its maximum when x and y are equal. Therefore, we want x and y to be as close to each other as possible.

We can set x = y to simplify the equation:

x + x = 25
2x = 25
x = 25/2
x = 12.5

Since x and y need to be non-negative, we can round down x to the nearest whole number, which is 12. Therefore, if x = 12, then:

y = 25 - x
y = 25 - 12
y = 13

So, the two non-negative numbers whose sum is 25 and whose product is as large as possible are 12 and 13.