Finding maxima and minima of $f(x,y) = x^{4} + y^{4} - x^{3}$ Finding maxima and minima of $$f(x,y) = x^{4} + y^{4} - x^{3}$$
I tried solving this question but need help.
In this question, I calculated the partial derivatives of first and second order.
I found
$$f_{x} = 4x^{3} - 3x^{2}\\   f_{y}= 4y^{3} \\ f_{xx} = 12x^{2}-6x \\ f_{yy}= 0$$
On equating $$f_{x} = 0 \hspace{0.5cm}  and\hspace{0.5cm} f_{y} = 0 $$ I get the critical points as (0,0) and (0.75,0) but I am not able to understand whether these points constitute a maxima or minima because $$f_{xx}f_{yy} - f_{xy}f_{yx}=0$$
Can somebody please guide?

For the minimum, if $x\ge 1$ or $x\le 0$ we have $x^4-x^3+y^4=x^3(x-1)+y^4\ge 0$.  If $0<x<1$ we have by AM-GM
$$x^3(1-x)=27(x/3)^3(1-x)\le 27\left(\frac{3(x/3)+(1-x)}{4}\right)^4=27/256$$
so $x^4-x^3\ge -27/256$ and so $x^4-x^3+y^4\ge x^4-x^3\ge -27/256$.  The minimum value is reached at $x=3/4$, $y=0$.
For the maximum, clearly it doesn't exist.
