Rolle's theorem

Problem

Verify the hypotheses of Rolle’s theorem for $f(x)=x^2-4x+3$ on $[1,3]$ and find $c$.

Solution

Hypotheses: $f$ is a polynomial, hence continuous on $[1,3]$ and differentiable on $(1,3)$. At the endpoints: $f(1)=1-4+3=0$ and $f(3)=9-12+3=0$, so $f(1)=f(3)$. The hypotheses of Rolle are satisfied.

Conclusion: there exists $c\in(1,3)$ with $f'(c)=0$. From $f'(x)=2x-4=0$ we obtain $\boxed{c=2}$, which belongs to $(1,3)$: it is the vertex of the parabola, where the tangent is horizontal.

Links

Topics: Derivatives
Concepts: Stationary point · Rolle’s theorem
Skills: Prove
People: Michel Rolle
Exercise type: Proof