Let $f\left(x\right)$ be a function such that

• $f\left(x\right)$ is continuous on $[𝒶,𝒷]$
• $f\left(x\right)$ is differentiable on $(𝒶,𝒷)$
• $f\left(𝒶\right)=f\left(𝒷\right)$

then there is $\omega \in (𝒶,𝒷)$ such that $f^{\prime }\left(\omega \right)=0$.

Let $f\left(x\right)$ be a function such that

• $f\left(x\right)$ is continuous on $[𝒶,𝒷]$
• $f\left(x\right)$ is differentiable on $(𝒶,𝒷)$

then there is $\omega \in (𝒶,𝒷)$ such that

If $f^{\prime }\left(x\right)=0$ for all $x\in (𝒶,𝒷)$ then $f\left(x\right)$ is constant on $(𝒶,𝒷)$.

If $f^{\prime }\left(x\right)=g^{\prime }\left(x\right)$ for all $x\in (𝒶,𝒷)$ then $f\left(x\right)-g\left(x\right)$ is constant on $(𝒶,𝒷)$ i.e.,

for some $𝒸\in ℝ$.