Let the function

$g(x,y)=-x^2+xy-\frac{1}{8}{y}^4$

whose first order partial derivatives are

$g_x=-2x+y,$ and $,g_y=x-\frac{1}{2}{y}^3.$

Determine the critical points and show how many of these points are relative maxima, relative minima, and saddle points. $($Answer from the smallest

$x-$values to the largest$).$

$(x,y)=(-\frac{1}{2},-1)$

$(x,y)=(0,0)$

$(x,y)=(\frac{1}{2},1)$

NOTE: ONLY $3$ ANSWER TRIES ON THE SECOND PART OF THIS PROBLEM BELOW.

saddle point

relative maxima

inconclusive

relative minima