Given, function is
f(x,y)=(1+xy)(x+y)=x+y+x2y+y2x Now, critical points are determined by ∇f(x,y)=0 , thus
∇f(x,y)=(1+2xy+y2,1+2xy+x2)=0⟹1+2xy+y2=0,1+2xy+x2=0⟹y2=x2⟹y=±x
Clearly, y=x as 1+3y2=0,∀y∈R
Thus, we conclude that y=−x⟹1−x2=0⟹x=±1
Therefore critical points are (1,-1) and (-1,1)
Now, we have to check the Hessian matrix for f at critical points to check whether above critical points are maxima, minima or saddle points.
Now,
Hf(x,y)=[2y2(x+y)2(x+y)2x] Now, clearly Hf(1,−1)=[−2002]&Hf(−1,1)=[200−2]
Clearly, both Hf(1,−1)&Hf(−1,1) are diagonal hence their eigen values are ±2
Hence, both Hf(1,−1)&Hf(−1,1) are semi positive definite ,thus (1,-1) and (-1,1) are saddle points.
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