Question

find the critical points of f(x y) = 4+x^3+y^3 -3xy help me plz

Answer 1

The critical point of a funtion in 3-D space is where the tangent plane is horizontal.

Perceive that the slopes of the partial derivatives at this critical point are both zero (they are both horizontal).

So, let's call fy the partial derivative of f with respect to y.
fx the partial derivative of f with respect to x.

fy = 3y^2 -3x
fx = 3x^2 -3y

Sum the two equations so you can simplify things : 3y^2 + 3x^2 -3x -3y = 0

3(y^2 + x^2 - x - y) = 0

y^2 + x^2 - x - y = 0

y^2 + x^2 = y + x

The unique numbers that squared and summed are equal to them summed are
(0,1), (1,0), (1,1), (0,0)

now you just have to plug in these numbers into the function and see when you get the smaller or the larger.

Ok ?