# Absolute Max/Min

Can anyone tell me the general procedure in doing the following procedure?

$$f(x,y)=xy^2$$ with domain $$x^2+y^2\leq4$$

Find it's absolute max & absolute min.

Okay, here is my thought procedure, tell me what I can fix.

So I would basically say, find the partial derivatives with respect to x and y and set them equal to zero.

$$f_x=y^2=0$$ $$f_y=2yx=0$$

so what's up? I plug that into the original equation? and then do the whole matrix thing to find if it's an absolute max or min? so point $$(x,y)=(0,0)$$

Plug into the matrix $$\left(\begin{array}{cc}f_x_x&f_x_y\\f_x_y&f_y_y\end{array}\right)$$

But I don't know how I would go about considering the $$x^2+y^2\leq4$$, do I find the boundary point? What are those? $$(x,y)=(2,0)=(0,2)=(-2,0)=(0,-2)$$ and then plug it into the original equation and then use

Plug into the matrix $$\left(\begin{array}{cc}f_x_x&f_x_y\\f_x_y&f_y_y\end{array}\right)$$

Am I on the right track? Can someone show me some guidance?

Related Calculus and Beyond Homework Help News on Phys.org
You use the Hessian matrix (what you did above) to classify critical points. But the extrema do not have to occur at critical points--they can also occur at the boundary. In this case, the boundary consists of a circle of radius 2. Think about how the function behaves on this circle...maybe rewrite in terms of angle and see what you find.

LCKurtz