Regarding this function $f(x,y)=1007x^2-x^{2014}+(e^y-1+2x^2)^2$.
I want to find the strict local minimum of $f$. I started calculating $\nabla$:
$\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$
$=(2014x+8x(2x^2+e^y-1)-2014x^{2013},2e^y(2x^2+e^y-1))^T$
Now, the gradient must be $\nabla f(x,y)=(0,0)^T$, so I have to set the two functions like this:
1) $2014x+8x(2x^2+e^y-1)-2014x^{2013}=0$
2) $2e^y(2x^2+e^y-1)=0$
Here's where I'm stuck. How do I solve the first equation for x, since I have $x^{2013}$?