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Question Number 133191 by pticantor last updated on 19/Feb/21
g(x,y)=x4+y4−2(x−y)2findcriticalspointsofg(x,y)andhersnature
Answered by Olaf last updated on 20/Feb/21
g(x,y)=x4+y4−2(x−y)2▽g=[∂g∂x∂g∂y]=[4x3−2(x−y)4y3+2(x−y)]▽g=[00]⇔(x−y)=2x3=−2y3⇔(x,y)=(0,0):thisistheonlycriticalpoint.Hessianmatrix:[∂2g∂x2∂2g∂x∂y∂2g∂y∂x∂2g∂y2]=[12x2−2x2212y2−2y]Hessianmatrixin(0,0):Hess(g)∣(0,0)=[0220]det(Hess(g)∣(0,0))=|0220|=−4det(Hess(g)∣(0,0))<0⇒(0,0)isasaddlepoint
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