The sum of two numbers is 70. Find the numbers if their product is a maximum.
Show steps please!
Let $\displaystyle x$ and $\displaystyle y$ be positive numbers.
Thus, $\displaystyle x+y=70$
However, their product $\displaystyle P=xy$ must be a maximum.
Since $\displaystyle x+y=70\implies x=70-y$
Thus, we see that $\displaystyle P=(70-y)y\implies P=70y-y^2$
Since it has to be a maximum, differentiate and set equal to zero.
$\displaystyle P'=70-2y\implies 70=2y\implies \color{red}\boxed{y=35}$
Now that we found $\displaystyle y$, we can find $\displaystyle x$:
$\displaystyle x+35=70\implies \color{red}\boxed{x=35}$
I hope this makes sense!
--Chris