Show that the Second Derivative Test $is$ inconclusive when applied $to$ the following function $at(0,0).$ Describe the behavior $of$ the function $at$ the critical point.

$f(x,y)=2x^{2}y-3$

Confirm that the function $f$ meets the conditions $of$ the Second Derivative Test $by$ finding $f_x(0,0),f_y(0,0),$ and the second partial derivatives $off.$

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

$f_{}(x,y)=,f_{yy}(x,y)=,f_{xy}(x,y)=$

Thus the function $f$ meets the conditions $of$ the Second Derivative Test. Next, find the discriminant.

$D(x,y)=$

Why $is$ the Second Derivative Test inconclusive?

$A.$ because $D(0,0)=0$

$B.$ because $f_{yy}(x,y)=0$

$C.$ because $(0,0)is$ not a critical point $off$

$D.$ because $D(0,0)>0$ and $f_{}(0,0)=0$

$E.$ because $f_x(0,0)=f_y(0,0)=0$

What $is$ the behavior $offat(0,0)?$ Choose the correct answer below.

$A.$ The function $f$ has a local minimum $at(0,0)$ because $f(x,y)>f(0,0)$ for all points $(x,y)in$ some open disk centered $at(0,0).$

$B.$ The function $f$ has a saddle point $at(0,0)$ because any open disk centered $at(0,0)x,yf(x,y)>f(0,0)x,y$ such that $f(x,y).$