Hello,

I'm not sure how to get started on the following: Create a function $\displaystyle F:\mathbb{R}^3 \to \mathbb{R}$, whose set of critical points is all of the twisted cubic, $\displaystyle c(t)=(t,t^2,t^3)$?

Thanks in advance!

Printable View

- Oct 23rd 2011, 05:40 PMkeityoCreate a function with certain critical points
Hello,

I'm not sure how to get started on the following: Create a function $\displaystyle F:\mathbb{R}^3 \to \mathbb{R}$, whose set of critical points is all of the twisted cubic, $\displaystyle c(t)=(t,t^2,t^3)$?

Thanks in advance! - Nov 4th 2011, 10:07 AMxxp9Re: Create a function with certain critical points
$\displaystyle F(x,y,z)=(y-x^2)^2+(z-x^3)^2$

The idea is, divide $\displaystyle R^3$ as $\displaystyle R^3 = R^2$ × c, so that we can project every point p of $\displaystyle R^3$ to $\displaystyle R^2$, with c to a single point (0,0). Then define a function like $\displaystyle u^2+v^2$ on $\displaystyle R^2$ which has a single critical point at (0, 0).