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

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To find the critical points of the function f(x, y) = x^4 + 4xy + y^4, the partial derivatives are calculated as f'x = 4x^3 - 4y and f'y = 4y^3 - 4x. Setting these derivatives to zero leads to the relationship y = x^3. The user initially believed this resulted in infinite critical points but later recognized that the equations x = y^3 and y = x^3 are not equivalent. The discussion highlights the importance of correctly interpreting the relationships between variables when identifying critical points.
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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?
 
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Nevermind, my brain is kinda tired... just realized that x = y^3 and y = x^3 is not the same equation. Well thanks anyway :).
 

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