Exercise 18Calculus › Single-variable calculus › Differentiation › Mean Value TheoremHigh schoolAdded 4 Nov 2026

Rolle's theorem on a polynomial

#Rolle#polynomial#existence

Statement

Let $P(x)=x^3-3x+1$ . Show that $P'$ has at least one root in $]-1,2[$ .

Hint

Check $P(-1)=P(2)$ and apply Rolle's theorem.

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Solution

Values. $P(-1)=-1+3+1=3$ and $P(2)=8-6+1=-1$ .

Apply Rolle on $[-1,2]$ . $P$ is a polynomial, continuous on $[-1,2]$ , differentiable on $]-1,2[$ , and $P(-1)=P(2)=3$ . By Rolle, there exists $c\in]-1,2[$ with $P'(c)=0$ .

Conclusion. $P'$ vanishes on $]-1,2[$ .

Rolle's theorem on a polynomial

Statement

Solution

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