Find and classify critical points of f ( x , y ) = cos ⁡ ( x 2 + y 2 )I have to classify the critical points of f, but the problem says to only consider the points ( 0 , 0 ) , ( π 2 , π 2 ) So, long story short, after using the second derivative test I got that D = 0 at both critical points.How should I proceed with this problem?

Question

Find and classify critical points of      f  (  x  ,  y  )  =  cos  ⁡  (      x    2    +      y    2    )I have to classify the critical points of f, but the problem says to only consider the points      (  0  ,  0  )  ,      (                            π                2              ,                            π                2              )  So, long story short, after using the second derivative test I got that      D  =  0 at both critical points.How should I proceed with this problem?

Anabella Moss · Accepted Answer

At (0,0), the function has a local maximum, because f(0,0)=1 and f(x,y)&amp;lt;1 if (x,y) is near to (but distinct from) (0,0).At       (                  π        2              ,                  π        2              )  , f has a local minimum, although not a strict one. In fact   f  (  x  ,  y  )  =  −  1 at every point of the circle       x    2    +      y    2    =  π (to which the point       (                  π        2              ,                  π        2              )   belongs) and   f  (  x  ,  y  )  &amp;gt;  −  1 at all other points near       (                  π        2              ,                  π        2              )

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