find all critical points of the function fxy 2yx423y332x ind

find all critical points of the function f(x,y) =2yx^4+(2/3y)^3-32x. indicate whether each such point gives a local maximum, a local minimum , or whether it is a saddle point.

f(x,y)=2yx^4+(2/3y)^3-32x.

f_x(x,y)=8*y*x^3-32

f_xx(x,y)=24*y*x^2

f_y(x,y)=2*x^4-(8/9)*(1/(y^4))

f_xy(x,y)=8*x^3

now critical point:-

f_x(x,y)2*x^4-(8/9)*(1/(y^4))=0 ==>x^4 * y^4=0

and

f_y(x,y)=2*x^4+2*y^2=0 ==>x^4+y^2=0

only possible if (x,y)=(0,0)

at (0,0)

f_xx(x,y)=24*y*x^2=0

f_xy(x,y)=8*x^3=0

so D=f_xx*f_yy-f_xy^2=0

hence no conclusion can be done

find all critical points of the function f(x,y) =2yx^4+(2/3y)^3-32x. indicate whether each such point gives a local maximum, a local minimum , or whether it is

find all critical points of the function fxy 2yx423y332x ind

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