This is the given function f ( x , y ) = x 4 − 2 ( x 2 ) ( y 2 ) + y 4 + y 3 how determine the nature of critical point (0,0) ?

Question

This is the given function  f  (  x  ,  y  )  =      x    4    −  2  (      x    2    )  (      y    2    )  +      y    4    +      y    3  how determine the nature of critical point (0,0) ?

Liehm1mm · Accepted Answer

The most straightforward way to answer these kinds of questions is to use the Hessian.  H  (  x  ,  y  )  =      [                            12                      x            2                    −          4                      y            2                                    −          8          x          y                                      −          8          x          y                          −          4                      x            2                    +          12                      y            2                    +          6          y                      ]  You will notice that at   (  0  ,  0  ), the Hessian is zero and the Hessian test is inconclusive about whether the critical point is a min/max or saddle point.We can try to evaluate   f  (  x  ,  y  ) for a specific slice and determine the behavior of that. So let&#039;s calculate it for   y  =  x.  f  (  x  ,  x  )  =  48      x    3  Along this slice, we see that   (  0  ,  0  ) is an inflection point. Note that any point which is an extrema would stay an extrema when looking along any direction. So if a point is not an extrema along a specific direction, then it cannot be an extrema of the original function.Thus,   (  0  ,  0  ) is a saddle point.

This is the given function f(x,y)=x^4−2(x^2)(y^2)+y^4+y^3 how determine the nature of critical point (0,0) ?

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