- #1
countzander
- 17
- 0
Find the global max/min for z=xy^2 - 5 on the region bounded by y=x and y=1-x^2 in the xy-plane.
The global maximum for z in this region is 0, which occurs at the point (0,0). The global minimum for z in this region is -5, which occurs along the line y=x.
To find the global maximum/minimum for z, we first need to find any critical points within the region. These are points where the partial derivatives of z with respect to x and y are both equal to 0. Then, we can evaluate the z values at these critical points and compare them to the z values along the boundary of the region to determine the global maximum/minimum.
Yes, the global maximum/minimum can occur at a point on the boundary of the region. In fact, in this case, the global minimum occurs along the line y=x, which is a boundary of the given region.
No, the global maximum/minimum is not unique in this region. In this case, the global minimum occurs along the line y=x, and there is another local minimum at the point (0,0).
The shape of the region can greatly affect the location of the global maximum/minimum. In this case, the region is bounded by the line y=x, which contains a local minimum for z. If the region were extended beyond this line, the global minimum would no longer be at this point.