How do I find the critical points of this multivariable function?

[Inertigratus]

Homework Statement

The function is defined in the xy-plane. Find all critical points and determine their characteristics.

Homework Equations

f(x, y) = x⁴ + 4xy + y⁴

The Attempt at a Solution

I tried finding the partial derivatives.
f'_x = 4x³ - 4y
f'_y = 4y³ - 4x
Setting them to 0 gives y = x³.
If x = 1, y = 1, if x = 2, y = 8, etc.
This means that there's an infinite amount of critical points, this can't be right?

 

[Inertigratus]

Nevermind, my brain is kinda tired... just realized that x = y^3 and y = x^3 is not the same equation. Well thanks anyway :).