Polynomials of odd degree can have no critical point.

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SUMMARY

This discussion centers on the properties of polynomials with real coefficients, specifically addressing critical points and roots. It is established that polynomials of odd degree must have at least one real root, as supported by the Intermediate Value Theorem. However, it is clarified that polynomials of odd degree can exist without critical points, exemplified by the polynomial \(y = \frac{x^3}{3} - x\), which has critical points at \(x = 1\) and \(x = -1\). The assertion that all even degree polynomials have at least one critical point is confirmed as true.

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batballbat
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Are these assertions true?
I am referring to polynomials with real coefficients.
1. There exists of polynomial of any even degree such that it has no real roots.
2. Polynomials of odd degree have atleast one real root
which implies that polynomial of even degree has atleast one critical point.
3. Polynomials of odd degree can have no critical point.
 
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The first two are. What do you mean by "critical point"? If you mean a place where the derivative is zero, then this is not right.

For 1) a polynomial of degree two that has no real roots is x^2 +1. Do you see how to generalise? For 2) you want to use the Intermediate Value Theorem.
 


2) not true: example y=x3 + x. It has a real root at x=0, but the derivative is never zero (for real x).
3) not true: example y=x3/3 - x. Critical points at x = 1, -1.
 


Did you edit #2 from the original post to add the second part? Because I didn't mean to include that part in my response. I was only saying that the first line of #2 is correct.
 


i don't get it.
Shouldnt a polynomial of even degree have atleast one critical point?

in question 3. i meant for any odd degree we can find a polynomial which has no critical point.
 


batballbat said:
i don't get it.
Shouldnt a polynomial of even degree have atleast one critical point?

in question 3. i meant for any odd degree we can find a polynomial which has no critical point.


All of them are true:

(1) \,(x^2+1)^n\,

(2) The intermediate value theorem for continuous functions

(3) \,\int (x^2+1)^ndx\,

DonAntonio
 


batballbat said:
i don't get it.
Shouldnt a polynomial of even degree have atleast one critical point?

in question 3. i meant for any odd degree we can find a polynomial which has no critical point.

Yes, it does, apparently I have misread your post twice now. If all odd polys. have a real root, then this means that the derivative of all even degree polys. have a zero, which means all even degree polys. have a critical point.
 

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