Given function $u = x^4 + y^4$ with extra condition $(x-1)^3-y^2=0$. Find extreme values of this function under this condition.

Question

Obvious way to solve it is to substitute $y^2=(x-1)^3$ right away. However, after that I am stuck with solving equation $2x^3 + 3(x-1)^5=0$, which for some reason occurred difficult to me.

So, I thought that maybe using Lagrange method could help. But in fact, in this case it led me to the same equation.

So, my questions are:

1. Is there any simple way to solve this equation?
2. Is it a coincidence that in this case Lagrange method didn’t give me anything useful? Or I could see this in advance?

Thanks for your help and answers in advance!