The sum of $3$ and $7$ is $10~ (3 + 7 = 10)$ and the product of $3$ and $7$ is $21~ (3 \times 7 = 21)$. Similarly, $2.1 + 3.2 + 4.7 = 10$ and $2.1 \times 3.2 \times 4.7 = 31.584$.
What is the largest product that can be made from numbers that sum to $10$?
Clearly, this problem is solvable with basic calculus but there should be an easier solution (this was asked in a high school math competition) but I can't find it. Thanks in advance.
On
Any number $n$ that is $5$ or larger can be replaced by $3$ and $n-3$, so there are no numbers larger than $4$. You might as well replace $4$ by $2,2$, so you will only have $2$s and $3$s. As $3,3$ beats $2,2,2$ you should have zero, one, or two $2$s. For the specific case of $10$, you do best with $2,2,3,3$ with product $36$
Hint: Using AM-GM, for $n$ terms the maximum is when each term is the same, to get a product of $(10/n)^n$.
This in turn is maximised when $n=4$.. to get $2.5^4= 39.0625$, the function is unimodal, so you could use calculus and try integers nearest to the optimum real number. Or you could track ratios of successive terms and find the maximum without calculus - that would be more cumbersome though.