For the following function $F(x,y)$, determine whether the set $S = {(x,y): F(x,y)=0}$ is a smooth curve. Draw a sketch of $S$ near any points where $\nabla F = 0$. Near which points of $S$ is $S$ the graph of a function $y = f(x)? x = f(y)?$
$$F(x,y) = (x^2 + y^2)(y - x^2 - 1) = 0$$
So based on the implicit function theorem, if $\nabla F \neq 0$ then it means i should be able to represent my fucntion in graphical form. So i've done a couple of these and i am doing a few things wrong according to the solutions and i don't understand why.
According to the solution for this question this fucntion is a point and a parabola, $\nabla F$ only equals $0$ at the point $(0,0)$, but not on the parabola, $y= f(x)$ on the parabola, $x = f(y)$ near any point on the parabola except $(0,1)$
Here are my questions:
- how do i find the points where $\nabla F=0$? I mean by taking the gradient of this function by inspection i could see $(0,0)$ would be a point, but what about if there are other points present?
- how can you tell that $F$ does not equal $(0,0)$ on the parabola? Aren't the two functions in $F$ multiplied together so how is it possible to distinguish?
3) with deciphering that you can represent the parabola as $x = f(y)$ almost everywhere except for at $(0,1)$, i noticed that this is the $x$-intercept and in all of other questions it also came about that i cannot solve for the graph of a function be it $y = f(x)$ or $ x = f(y)$ at the intercept how does the intercept play a role?
A lot here i know but thanks