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Thread: Create a function with certain critical points

  1. #1
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    Create 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!
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  2. #2
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    Re: 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).
    Last edited by xxp9; Nov 4th 2011 at 10:23 AM.
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