Prove Theorem 1 in the lesson 'Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions'.

Problem: 

Assume that $f(x)$ is differentiable on an open interval $I$. Prove that

  1. If $f\,'(x)\gt 0$ for all $x$ in $I$ then $f(x)$ increases on $I$.
  2. If $f\,'(x)\lt 0$ for all $x$ in $I$ then $f(x)$ decreases on $I$.
  3. If $f\,'(x)=0$ for all $x$ in $I$ then $f(x)$ is constant on $I$.
Answer: