Question

Find two positive integers whose sum is 60 and whose product yields a maximum

Answer 1

So you want \(a+b=60\) and \(a\cdot b=y\) is maximized. First, rearrange so that you have \(b=60-a\) and substitute to get\[y=a\cdot(60-a)=60a-a^2\]You want to find the maximum of that last equation.

Answer 2

To do that, take the derivative.\[y'=60-2a\]Set it equal to 0, and solve for a.\[0=60-2a\implies a=30\]Now you need to find what \(b\) was. Fortunately, we have the relation \(a+b=60\). So \[30+b=60 \implies b=30\]