The following question is incomprehensible to me: ''Prove that for the equation $e^{y-1} +\ln y +x^3 =1$ there exists a differnetiable solution $y(x)$ in the neighbourhood of $ x_0=0$ and fulfills $y(0)=1$. Prove that $y'(0)=y''(0)=0$. Sort the extremum point $x=0$ of $y(x)$.''
When do they mean when they say '' there exists a differentiable solution''? Isn't it true that for every equation of continuous and differentiable functions, the solution/s will always be differentiable either way?
HINT.-By "solution "$y(x)"$ you must understand that there is in the plane curve of equation $F(x,y)=e^{y-1} +\ln y +x^3-1=0$ an implicit function $y=f(x)=\text{?}$ defined in a certain neighbourhood of $x_0=0$ about which you must proved the requested properties. I feel that you cannot find a closed form for $f(x)$ but it is possible to prove the required questions.