Let U be some domain in R^n with 0 in U. We have some C^2 function f:U->R with grad f(0)=0, therefore 0 is a critical point (i.e. local maximum, local minimum or saddle point). Define grad f(0)=0 and f^+:U^+->R,x|->f(x). Is 0 an extremum of f^+ and if yes how could we prove it?

charmbraqdy 2022-11-21 Answered
Let U be some domain in R n with 0 U. We have some C 2 function f : U R with f ( 0 ) = 0 , therefore 0 is a critical point (i.e. local maximum, local minimum or saddle point). Define f ( 0 ) = 0 and f + : U + R , x f ( x ).
Is 0 an extremum of f + and if yes how could we prove it?
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Answers (1)

zastenjkcy
Answered 2022-11-22 Author has 14 answers
No. As a counterexample, take a function f 0 : R R with a saddle at the origin (e. g. f 0 ( x ) = x 3 ) and then
f : R 2 R , f ( x , y ) = f 0 ( x ) ,
its constant extension in y-direction. Then the point 0 is a critical point of f and its cutoff f + to non-negative second coordinate { y 0 } still does not have an extremum in ( x , y ) = ( 0 , 0 ) (because f ( x , 0 ) = f 0 ( x ) has a saddle in x = 0).
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