Find all critical points of the function fxy 16y 13x3 xy4

Find all critical points of the function f(x,y) = 16y - (1/3(x^3)) - x(y^4) +5.

Indicate whether each such point gives a local maximum, a local minimum, or whether it is a saddle point.

Solution

df/dx=0

x^2 -y^3=0
x^2=y^3

df/dy=0
16-4xy^3=0

16=4xy^3=4x^3

4=x^3
x=4^(1/3)
y^3=x^2=4^(2/3)
y=4^(2/9)

So the critical point is x₀=4^(1/3)
y₀=4^(2/9)

A=d²f/dx²=2x>0

B=d²f/dxdy=-3y²

C=d²f/dy²=-12xy^2

B^2-AC=9y^4-24x^2y^2=3y^2(3y^2-8x^2)<0

for point (x₀,y₀) so it is a local minimum value