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?

Question

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?

zastenjkcy · Accepted Answer

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).

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?

Answered question

Answer & Explanation

New Questions in Differential Calculus