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This is a question from Cheney and Kincaid's Numerical Analysis (3rd ed, pg 138).

Consider the homotopy $h(t,x)=tf(X)+(1-t)g(x)$, in which $f(x)=x^2-5x+6$ and $g(x)=x^2-1$. Show that there is no path connecting a root of $g$ to a root of $f$.

I'm a bit confused, since clearly you can create a homotopy between them. The chapter text does not provide any insight. Are there other requirements to connect the roots besides there being a homotopy?

Georgia S
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  • I think that you should clarify what do they exactly mean by the "path connecting root". It's clear that there is a homotopy, but it's possible that somewhere in between you have a polynomial with no real roots. And that's an obstruction for path existence. – Evgeny Oct 07 '15 at 17:38
  • @Evgeny, Oh, so if I can find a t such that h(t, x) does not have roots (and thus I can't find a solution at that t), then a path doesn't exist? – Georgia S Oct 07 '15 at 17:44
  • Yep, that's exactly what I've meant :) – Evgeny Oct 07 '15 at 17:45

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