# Extrema problem

1. Jul 17, 2007

### kekido

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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:
$$H=\left(\begin{array}{cc}0&-1\\-1&0\end{array}\right)$$
$$Q(x,y)=Hk\cdotk=-2xy$$
$$Q(a)=Q(0,1)=0$$

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.

2. Jul 17, 2007