What type of critical point is ( 0 , 0 ) in the following function? f ( x , y ) ∈ C 2 , T 2 ( x , y ) = 3 + 2 x + 0.5 x 2 − x y − y 2

Question

What type of critical point is   (  0  ,  0  ) in the following function?  f  (  x  ,  y  )  ∈      C    2    ,      T    2    (  x  ,  y  )  =  3  +  2  x  +  0.5      x    2    −  x  y  −      y    2

Krystal Dillon · Accepted Answer

Assuming that       T    2   is the Taylor expansion up to degree   2 around   (  0  ,  0  ), it follows that   ∇  f  (  0  ,  0  )  =  (  2  ,  0  ) so it is NOT a critical point.Recall that       T    2    (  x  ,  y  )  =  f  (  0  ,  0  )  +  ∇  f  (  0  ,  0  )  ⋅  (  x  ,  y  )  +  (  x  ,  y  )      H    f    (  0  ,  0  )      (                            x                                      y                      )

What type of critical point is (0,0) in the following function?f(x,y) in C^2,T_2(x,y)=3+2x+0.5x^2−xy−y^2

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