I am trying to find the absolute maxima and minima of
$ f(x,y) = 2x^3 + y^4$ in the domain $D = \{(x,y)| x^2 + y^2 \le 1 \}$

I have made an attempt as shown in attached picture. Please ignore dashed areas.
[Attached picture](https://i.stack.imgur.com/I0yc5.jpg).
The answer is (1,0) and (-1,0). I do NOT want to use the Lagrange multiplier method to solve.

Is it legal to substitute $y^4 = (1-x^2)^2$ into $f(x,y)$ to make a single variable $f(x)$ and then find $f'(x) =0 $? Then, after finding the $x$ can I substitute into $x^2 +y^2 = 1$ to find y? Then can I plug it back into $f(x,y)$ to find critical points?

I want to know how to find max and min on the boundary $x^2 + y^2 = 1$ without the LM method?

Thanks for your help!