Mean Value Theorem

For Rolle's Theorem to be applicable on $[a,b]$, the following conditions must be satisfied:

1. $f(x)$ is continuous on the closed interval $[a,b]$.
2. $f(x)$ is differentiable on the open interval $(a,b)$.
3. The function values at endpoints must be equal: $f(a)=f(b)$.

• Conclusion: If these hold, there exists at least one $c\in (a,b)$ such that $f^{\prime }(c)=0$ (the tangent is horizontal at $c$).