Exercise 1276Calculus › Single-variable calculus › Differentiation › Mean Value TheoremL1Added 4 Nov 2026

Uniqueness of a root

#Rolle#uniqueness#contrapositive

Statement

Let $f(x)=x^3+x+1$ . Show that $f$ has exactly one real root.

Hint

Use the IVT for existence; use Rolle for uniqueness.

Show solutionHide solution

Existence. $f(-1)=-1<0$ , $f(0)=1>0$ ; IVT gives a root in $]-1,0[$ .

Uniqueness. If there were two roots $a<b$ , Rolle gives $c\in]a,b[$ with $f'(c)=0$ . But $f'(x)=3x^2+1>0$ for all $x$ : contradiction.

Conclusion. $f$ has exactly one real root.