Thread: Create a function with certain critical points

1. Create a function with certain critical points

Hello,

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

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