The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $I$ be an interval. Call $x\in I$ an inner point of $I$ if $u<x<v$ for some $u,v\in I$. Let $f:I\to\mathbb{R}$ be continuous on $I$ and differentiable at every inner point of $I$, derivatives being taken relative to $I$.

(i) If $0\le f'(x)$ for every inner point $x$ of $I$, then $f(x)\le f(y)$ for all $x,y\in I$ with $x\le y$.

(ii) If $0<f'(x)$ for every inner point $x$ of $I$, then $f(x)<f(y)$ for all $x,y\in I$ with $x<y$.

(iii) If $f'(x)\le0$ for every inner point $x$ of $I$, then $f(y)\le f(x)$ for all $x,y\in I$ with $x\le y$.

(iv) If $f'(x)<0$ for every inner point $x$ of $I$, then $f(y)<f(x)$ for all $x,y\in I$ with $x<y$.

(v) If $f'(x)=0$ for every inner point $x$ of $I$, then $f(x)=f(y)$ for all $x,y\in I$.

(vi) If $M\in\mathbb{R}$ and $|f'(x)|\le M$ for every inner point $x$ of $I$, then $|f(y)-f(x)|\le M\,|y-x|$ for all $x,y\in I$.