I have to describe the behaviour, while $c$ is changing, of the level curve $f(x,y)=c$ for the function $f(x,y)=x^3-x$.
I have done the following:
The level curves are defined by $$\{(x,y)\mid x^3-x=c\}$$
For $c=0$ we have that the set consists of the lines $x=0,x=1,x=-1$.
Is it correct so far?
How could we continue? What can we say about the other values if $c$? Which is the set when $c$ is positive and which when $c$ is negative?
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EDIT:
According to Travis' answer we have that any level set $\{f(x, y) = c\}$ is a union of the vertical lines $\{x = x_0\}$ in the plane, where $x_0$ varies over the roots of $f(x, y) -c = x^3 - x + c$.
Since this is a cubic polynomial, depending on the value of $c$ it can have three real single roots, one real single root and one real double root, or one real single root and two nonreal roots.
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Why do we describe in that way the behaviour of the level curve?
When the polynomial has three roots does the level curve consist of three lines?
What happens when the polynomial has a double root?
And what happens when it has non-real roots?
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Also what information do we get for the graph of $f$ ?
The function $f(x, y)$ depends only on $x$, so if $(x_0, y_0)$ is on the level curve, so is $(x_0, y)$ for every $y \in \mathbb{R}$. Thus, any level set $\{f(x, y) = c\}$ is a union of the vertical lines $\{x = x_0\}$ in the plane, where $x_0$ varies over the roots of $$f(x, y) = x^3 - x - c$$ regarded as a function of $x$ alone. Since this is a cubic polynomial in $x$, depending on the value of $c$ it can have three real single roots, one real single root and one real double root, or one real single root and two nonreal roots.