Essential Concepts

• If $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$ and $f(a)=0=f(b)$, then there exists a point $c \in (a,b)$ such that $f^{\prime}(c)=0$. This is Rolle’s theorem.
• If $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$, then there exists a point $c \in (a,b)$ such that
  $f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}$.

  This is the Mean Value Theorem.

• If $f^{\prime}(x)=0$ over an interval $I$, then $f$ is constant over $I$.
• If two differentiable functions $f$ and $g$ satisfy $f^{\prime}(x)=g^{\prime}(x)$ over $I$, then $f(x)=g(x)+C$ for some constant $C$.
• If $f^{\prime}(x)>0$ over an interval $I$, then $f$ is increasing over $I$. If $f^{\prime}(x)<0$ over $I$, then $f$ is decreasing over $I$.