Question

Find two integers whose sum is 26 and whose product is maximum.

Answer 1

make them equal by symmetry

Answer 2

What do do you mean?

Answer 3

suppose i call one integer \(x\) and the other integer \(y\) then you have
\[x+y=26\] or
\[y=36-x\] and you want to maximize
\[xy=x(26-x)\]

Answer 4

typo there, i meant \(y=26-x\) but you get the idea

Answer 5

it is easy enough to use algebra to maximize
\[A(x)=x(26-x)=26x-x^2\] to find the vertex of the parabola
but it is easier to think as follows

Answer 6

I think I understand. So what does the maximum mean in the problem?

Answer 7

find the value of \(x\) that makes
\[26x-x^2\] as large as possible
i.e. find the vertex of the parabola

Answer 8

but as i said, it is a lot easier than doing that
since you cannot tell x from y in this question, you get the same answer if you replace y by x
therefore they must be equal, i.e. they are both 13