Finding & Classifying Local/Absolute Extrema of f(x,y)

[kekido]

Homework Statement

Find and classify the local and absolute extrema of the function
$$f(x,y)=x-xy$$ over the region
$$D={(x,y)|x^2+y^2\leq1 and x+y\geq0}$$

Homework Equations

The Attempt at a Solution

Critical points are where the first derivative (gradient) is 0.
$$\nablaf=(1-y, -x)=0$$
So critical point a=(0,1)

In order to classify the critical point, find the Hessian matrix of f at a:
$$H=\left(\begin{array}{cc}0&-1\\-1&0\end{array}\right)$$
Then the quadratic form is:
$$Q(x,y)=Hk\cdotk=-2xy$$
$$Q(a)=Q(0,1)=0$$

Which means the test is inconclusive? I.e., the critical point a is a saddle point, which is neither local maximum or minimum. Am I right?


Also, how do you find the absolute extreme of the function on the region aforementioned?

I tried to convert x^2+y^2<=1 to polar coordinates, which gives
$$r^2\cos^2\eta+r^2\sin^2\eta\leq1$$
$$r^2\leq1$$
$$0<r\leq1$$
However, this doesn't help much as the original function converted to polar coordinates is not straightforward to find its extrema given the domain of r and theta.

 

[kekido]

Oops...Sorry for the double post. I was editing my original post but somehow it got reposted...Please ignore this one and use the other:

https://www.physicsforums.com/showthread.php?t=177299

Thanks.