"Let $f:\mathbb R^n\to\mathbb R^n$ be any differentiable function from $\mathbb R^n$ to itself. Let $x_0$ be a point in $\mathbb R^n$ with $f(x_0)=0$ but with $\det(Df(x_0))\neq0$, where $Df$ denotes the Jacobian of the function $f$. Find a function $g:\mathbb R^n\to\mathbb R^n$ that has the point $x_0$ as a fixed point."

I'm completely lost. I don't even know what class this is for.

Question

"Let $f:\mathbb R^n\to\mathbb R^n$ be any differentiable function from $\mathbb R^n$ to itself. Let $x_0$ be a point in $\mathbb R^n$ with $f(x_0)=0$ but with $\det(Df(x_0))\neq0$, where $Df$ denotes the Jacobian of the function $f$. Find a function $g:\mathbb R^n\to\mathbb R^n$ that has the point $x_0$ as a fixed point."

I'm completely lost. I don't even know what class this is for.

anonymous · Answer

For reference, this question was in our school's GRE review/problem set.

lifeisadangerousgame · Answer

you lost me at "$$f:\ $$

anonymous · Answer

@Zarkon One of my issues with this problem is that I can't even identify the class I learned this in. I feel like each of the individual concepts I'm seeing here were picked up in different classes.

Can you identify the specific class?

anonymous · Answer

How about
$$
g(x) = x
$$
All points are fixed points including
$$
x_0
$$

anonymous · Answer

@eliassaab I guess I'm not as familiar with Jacobians as I thought, because I don't follow that answer. :(

anonymous · Answer

Did you copy your problem correctly?

anonymous · Answer

Yes. Word for word.

anonymous · Answer

@FoolForMath Can you help?

anonymous · Answer

Oh well. :( I give up.