f(x,y)=xye−x2−y2
fx=ye−x2−y2−2x2ye−x2−y2
fy=xe−x2−y2−2xy2e−x2−y2
fx=0fy=0=>ye−x2−y2−2x2ye−x2−y2=0xe−x2−y2−2xy2e−x2−y2=0
y(1−2x2)=0x(1−2y2)=0 Critical points:
(−22,−22),(−22,22),(0,0),
(22,−22),(22,22)
fxx=−6xye−x2−y2+4x3ye−x2−y2
fxy=e−x2−y2−2y2e−x2−y2−2x2e−x2−y2+4x2y2e−x2−y2
fyy=−6xye−x2−y2+4xy3e−x2−y2
(−22,−22)
fxx=−2e−1<0
fxy=0
fyy=−2e−1
∣∣−2e−100−2e−1∣∣=4e−2>0 Point (−22,−22) is a local maximum.
(−22,22)
fxx=2e−1>0
fxy=0
fyy=2e−1
∣∣2e−1002e−1∣∣=4e−2>0 Point (−22,22) is a local minimum.
(22,−22)
fxx=2e−1>0
fxy=0
fyy=2e−1
∣∣2e−1002e−1∣∣=4e−2>0 Point (22,−22) is a local minimum.
(22,22)
fxx=−2e−1<0
fxy=0
fyy=−2e−1
∣∣−2e−100−2e−1∣∣=4e−2>0 Point (22,22) is a local maximum.
(0,0)
fxx=0
fxy=1
fyy=0
∣∣0110∣∣=−1<0 Point (0,0) is a saddle point.
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