What does it mean when critical points have non-real answers?

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SUMMARY

The discussion centers on finding critical points and local maxima/minima for the function y = x/(x^2 - 1). The user correctly identifies that to find critical points, one must take the derivative and set it to zero. However, the resulting critical point x = sqrt(-1) indicates a non-real solution, suggesting that no real critical points exist for this function. Additionally, the user notes that the derivative approaches zero at infinity and identifies vertical asymptotes at x = ±1.

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Haroldoo
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Hi, I was wondering if I could get an answer to a question that has been bothering me all day today.

Ok, so I got an assignment today and it wants me to find the Critical X's and Local Max/Min points for the equation: y = x/x^2-1

Correct me if I'm wrong on any of this. To find the critical X's I take the derivative and equate it to zero then solve for x to get the critical X's and from there I can find the Max/Min vaules.

After doing that I got x=sqrt(-1). Now I know that is a non real answer, so does that mean that those critical points do not exist? and does that also mean that there is no local max/min points? What exactly does it mean when I get the non real answers for my critical x values. Oh and I also found the critical x's when y is undefined to find the asymptotes.

If i wrote this like crap and you can't understand it, sorry heh, but any feedback is appreciated
 
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If you were to go to www.calc101.com and go to the graphing section, input your function and you will see that there are no points where the gradient is actually zero, although it does approach zero at plus and minus infinities. Other critical points include where the graph's derivative goes to infinity, at plus and minus 1.
 

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