So I am given a function of $x$ and $y$:
The partial derivative with respect to $x$, and the partial derivative with respect to $y$ are found. But there are a few things in the solution given that don't make sense to me.
Firstly, what is the thinking behind eliminating y (combining the results of both partial derivatives). Is this done due to the partial wrt $x$ representing all of the $x$ points where there is a gradient of zero, and the partial wrt to $y$ representing all of the $y$ points where the gradient is zero, so combining the results will give you the $(x, y)$ points of where the gradient is zero?
Secondly, once $y$ is eliminated to find the values of $x$ that satisfy both partial derivatives, how is the factorizing done. It's an easy cubic but what is the actual method for figuring out where to go from $x^4 = x$ ?
Also, once the factorizing is done, it says there are two $x$ points that satisfy the equation.
$x = 0$ due to the $x$ term being multiplied to everything ]
$x = 1$ due to the root of $(x - 1)$
but there is another factor, $(x^2 + x + 1)$. Is the reason there are no solutions from this factor due to the fact that if the quadratic formula is used to factorize this factor, we will end up with complex roots?