What is the minimum product of two numbers whose difference is 48? What are the numbers?
The minimum product is_____
The two number yield this product?______
Simplify your answer

Question

What is the minimum product of two numbers whose difference is 48? What are the numbers? 
The minimum product is_____
The two number yield this product?______
Simplify your answer

kinggeorge · Answer

You have two numbers $x, y$ such that $x-y=48$ and you want to minimize $xy$. First, rewrite the first equation to get $$x=48+y$$Then substitute that into the second equation. Thus, you get $$(y+48)y=y^2+48y$$Now we want to minimize that. Can you do that on your own?

anonymous · Answer

Need help please!

kinggeorge · Answer

You'll want to take the derivative of that function. So you get $2y+48$. To find the minimum, find the zero of this function. This results in $$2y+48=0 \implies y=-\,{48 \over 2}=-24$$That's the value for y that gives you the minimum value. Now we need to find x. But wait! we have a function that solves for x in terms of y already. $$x=48+y\implies x=48-24=24$$So we have that $$x=24$$$$y=-24$$ The product of these numbers is $$xy=24(-24)=-576$$