Finding Critical Points?

Question

how do you find the critical points of an equation if it has both x and y variables. For instance.

f(x,y) = 3x - (x^3) - 2(y^2)+(y^4)

I just don't know how to get the critical points because when you take the derivative and split it up into x and y components, there is no way to get the value of the other component.

Answer 1

f'_x (x,y)= 3-3x^2 (the partial derivative of f with respect to x)

f'_y(x,y)=-4y+4y^3 (the partial derivative of f with respect to y)

both have to be equal to 0,
so
3-3x^2=0, so x^2=1, x=1 or x=-1

Answer 2

You compute the gradient, which is a vector consisting of two components: df/dx and df/dy (these are partial derivatives with respect to x and y). Then you find the values of x and y for which BOTH df/dx and df/dy are equal to zero.

To compute df/dx, treat y as a constant and differentiate with respect to x:
df/dx = 3 - 3x^2

To compute df/dy, treat x as a constant and differentiate with respect to y:
df/dx = -4y + 4y^3

Now figure out when df/dx = 0:
3-3x^2 = 0 means 3=3x^2, or x^2=1, so x=1 or -1.

Figure out when df/dy = 0:
-4y + 4y^3 = 0 means 4y^3 = 4y, so either y=0, or y^2=1 (after dividing through by y), in which case y=1 or -1.

Therefore, the critical points are:
(1,0), (1,1), (1,-1), (-1,0), (-1,1), (-1,-1).