1. ## Optimisation

The sum of two numbers is 70. Find the numbers if their product is a maximum.

2. Originally Posted by bubbles73
The sum of two numbers is 70. Find the numbers if their product is a maximum.

Let $x$ and $y$ be positive numbers.

Thus, $x+y=70$

However, their product $P=xy$ must be a maximum.

Since $x+y=70\implies x=70-y$

Thus, we see that $P=(70-y)y\implies P=70y-y^2$

Since it has to be a maximum, differentiate and set equal to zero.

$P'=70-2y\implies 70=2y\implies \color{red}\boxed{y=35}$

Now that we found $y$, we can find $x$:

$x+35=70\implies \color{red}\boxed{x=35}$

I hope this makes sense!

--Chris