Explore Level Curves of $f(x,y)=x^3-x$

[mathmari]

Hey!

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?? (Wondering)

How could we continue ?? What can we say about the other values if $c$?? Which is the set when $c$ is positiv and which when $c$ is negative?? (Wondering)

 

[Fallen Angel]

Hi,

Plot $x^3-x$.

Geometrically, what you are doing is drawing a vertical line $x=whatever$ at the intersection points of an horizontal line $y=c$ with the curve $y=x^3-x$.

You will got as many vertical lines as intersection points you have with the horizontal one, taking a look at the plot you can easily figure out what happens, and then try to describe it analitically (if needed).

 

[mathmari]

I got stuck right now...

Do I not have to take cases for $c$ ($c=0$, $c<0$, $c>0$) and find which the set of level curves is described?? (Wondering)

Or am I supposed to do something else?? (Wondering)

 

[mathmari]

The level curves are defined by the set $L_c=\{(x,y)\mid x^3-x=c\}$.

The function $f(x,y)$ depends only on $x$. So, if $(x_0,y_0)\in L_c$, then $(x_0, y)\in L_c, \forall y\in \mathbb{R}$.

Therefore, each level set $\{f(x,y)=c\}$ is the union of the lines $\{x=x_0\}$ in the plane, where $x_0$ belongs to the set of roots of $f(x,y)-c=x^3-x-c$.

Since this a cubic polynomial, depending on the value of $c$, it can have three real simple roots, one real simple root and one real double root or one real simple root and two non-real roots, right? (Wondering)
Do we describe in that way the behaviour of the level curve? (Wondering) When the polynomial has three roots does the level curve consist of three lines? (Wondering)

What happens when the polynomial has a double root? (Wondering)

And what happens when it has non-real roots? (Wondering) Also what information do we get for the graph of $f$ ? (Wondering)