I am working on polynomial function, when it is possible for a polynomial function to have less real distinct root than its derivative function. I am curious whether any form or example of a cubic function with real coefficient that have only two real distinct roots, and its derivative function has only one real distinct root. Could you please give me an example of this condition? It would be very helpful. Thanks in advance.
$\endgroup$ 22 Answers
$\begingroup$Between any two (distinct) real roots of a cubic with real coefficients, its derivative has a real root, by Rolle's Theorem. Now the only way a cubic with real coefficients can have just two (distinct) real roots is if one of them is a double root. But if a cubic has a double root, that root is also a root of the derivative. Thus the derivative will have two distinct real roots.
$\endgroup$ 1 $\begingroup$Hints:
a cubic polynomial with real coefficients can have either $1$ or $3$ real roots, never $2$;
the derivative of a cubic is a quadratic, which can have either $2$ or $0$ real roots, never $1$;
look at cubics which have both a local maximum and a local minimum, but no real root in between them, such as for example $x^3-x+1$.
