In each case, for a) f has local maximum at (1,1) and for b) f has saddle point at (1,1).
a) f_{xx}f_{yy}-(f_{xy})2
=(-4)(-2)-(1)2
=8-1
=7>0
f_{xx}=-4<0
Therefore, f has local maximum at (1,1)
b) f_{xx}f_{yy}-(f_{xy})2
=(-4)(-2)-(3)2
=8-9
=-1<0
Therefore, f has saddle point at (1,1)
A factor at which a feature of variables has partial derivatives identical to 0 however at which the feature has neither a most nor a minimal value.
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Complete question:
Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In each case, what can you say about f?a) f_{xx}f_{yy}-(f_{xy})2 and b) f_{xx}f_{yy}-(f_{xy})2
