Exercise 1275Calculus › Single-variable calculus › Differentiation › Mean Value TheoremL1Added 4 Nov 2026

Rolle on a polynomial with three roots

#Rolle#multiple roots#derivative

Statement

Let $P$ have three distinct real roots. Show that $P'$ has at least two distinct real roots.

Hint

Apply Rolle between consecutive roots.

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Setup. Let $a<b<c$ be the roots. On $[a,b]$ , Rolle gives $c_1\in]a,b[$ with $P'(c_1)=0$ ; on $[b,c]$ , gives $c_2\in]b,c[$ with $P'(c_2)=0$ .

Conclusion. $c_1<c_2$ are two distinct roots of $P'$ .