Find the local maxima, local minima and saddle points?

Question

Find the local maxima, local minima and saddle points of the function f(x, y)=y^3-6xy+8x^2.

fx=-6y+16x

fy=3y^2-6x

-6y+16x=0

3y^2-6x=0

y=8x/3

3(8x/3)^2-6x=0

I'm stucked.

Answer 1

Oh! But you're so close!

3(8x/3)^2 - 6x = 0

3(64x^2/9) - 6x = 0

64x^2 / 3 - 6x = 0

64x^2 - 18x = 0

2x(32x - 9) = 0

x = 0, 9/32

y = 0, 3/4 respectively

so the two points are (0, 0) and (9/32, 3/4)

using the second derivative test we can find that (0, 0) is a saddle point and (9/32, 3/4) is a local minimum