Let f : R 2 → R be a function such that f ( x , y ) = 5 x 2 + x y 3 − 3 x 2 y .Is ( 0 , 0 ) a local maximum, local minimum or a saddle point?

Question

Let   f  :                    R            2        →            R       be a function such that  f  (  x  ,  y  )  =  5      x    2    +  x      y    3    −  3      x    2    y  .Is   (  0  ,  0  ) a local maximum, local minimum or a saddle point?

Laci Conrad · Accepted Answer

Informally, we only need to look at the function when   x and   y are small. In that case, the 3rd order element       x    2    y and the 4th order elements   x      y    3   can be neglected in front of the 2nd order element   5      x    2  . So   f  (  x  ,  y  )  ≈  5      x    2  , which looks like a parabolic cylinder in the Oy direction.More formally, replace   f by the first terms of a Taylor series :  f  (  x  ,  y  )  =  f  (  0  ,  0  )  +            ∂      f              ∂      x        x  +            ∂      f              ∂      y        y  +                    ∂        2            f              ∂              x        2                  x    2    +                    ∂        2            f              ∂      x      ∂      y        x  y  +                    ∂        2            f              ∂              y        2                  y    2    +  .  .  .  .keeping only the terms of the smallest degree, which gives you an approximation of the function in the neighborhood of   (  0  ,  0  ).In our case, your   f is a polynomial, so the Taylor expansion is itself.

Let f:R^2->R be a function such that f(x,y)=5x^2+xy^3−3x^2y.Is (0,0) a local maximum, local minimum or a saddle point?

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