How to Differentiate two equations to find Maximum Values I am stuck on this Differentiation problem, any help would be great!

If $A=xy$ and $x+5y=20$ find the maximum value of $A$ and the values of $x$ and $y$ for which this maximum value occurs


There is a theorem that the product of two numbers is greatest when they are equal, which can be proved in an elementary way using the identity $$4ab=(a+b)^2-(a-b)^2$$ when you clearly need $(a-b)^2$ to be as small as possible.
You can use this here by setting $z=5y$ so that $5A=xz$ and $x+z=20$: then $x=z=5y$ to give the maximum value of $5A$ and hence of $A$.
To do the same thing in a simple way with calculus, let $A=xy$ and $x+y=B$ be fixed. Then $y=B-x$ and $A=x(B-x)=Bx-x^2$.
Then $\frac {dA}{dx}=B-2x$ and the turning point is at $x=\frac B2$ (which agrees with the previous calculation).
You can do the calculus bit directly in your question by setting $x=20-5y$ and computing everything in terms of $y$. But I did want to show you some of the tricks you can use, because they are sometimes easier, and occasionally give a helpful check on a result obtained by other means.
