Finding maxima and minima of $f(x,y)=x^4+y^4-2x^2$

Your function can be written as $$f(x,y)=(x^2-1)^2+y^4-1$$ and we have $$(x^2-1)^2+y^4-1\geq -1$$. And we can conclude that there is no maximum.($\infty$)

You can analyze the function by completing the square.

Note that $$x^4-2x^2+y^4=(x^2-1)^2+y^4-1$$

The minimum of $-1$ occurs at $$(x^2-1)^2+y^4=0$$ That is $x=\pm 1,y=0$

The critical point $(0,0)$ is a saddle point because you get both positive and negative values near that point where the value of your function is $0$

There is no local or absolute maximum.