What does this clause in my question mean for finding maximum/minimum values?

Question

The question reads:

Find the maximum and the minimum of the function $f(x,y) = 4xy$ subject to $x^2 + y^2 = 8$.

I know to find the maximum and minimum, you must find the points where both $f_x \text{ and } f_y$ equal 0.

$f_x = 4y = 0 \implies y = 0$
$f_y = 4x = 0 \implies x = 0$

But now that I have this, I'm not sure what to do because I don't understand what is meant by the function being "subject" to $x^2 + y^2 = 8$.

Can someone please explain this clause and point me where to go next?

Answer 1

"Being subject to" means that the only values of x and y that you are allowed to put into your function $f(x,y)=4xy$ are values such that $x^2+y^2=8$. I would transform into polar coordinates because then the constraint is only on one variable and then see what you can do from there.

After converting to polar coordinates, the constraint becomes $r^2=8$ and $f(r,\theta)=4r^2\sin{\theta}\cos{\theta}=2r^2\sin{2\theta}$

Considering the function subject to the constraint $f(\sqrt{8},\theta)=16\sin{2\theta}$

Answer 2

There are good ways to solve this kind of constrained question, such as Lagrange multipliers, but if that is beyond you then you could note that $x^2+y^2=8$ implies $y=\pm \sqrt{8-x^2}$ and so you are trying to find values of $x$ where $4x\sqrt{8-x^2}$ is a maximum or minimum (or $-4x\sqrt{8-x^2}$ is, but this is much the same question) and the corresponding values of $y$.

Or you could take finagle29's suggestion, noting the constraint is a particular circle, and substitute $x=\sqrt8 \cos\theta$ and $y=\sqrt8 \sin\theta$ into $4xy$ and then find $\theta$ which maximises or minimises $4xy$.