Derivative sign implies monotonicity

Derivative sign implies monotonicity: Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$.

• If $f'(x)\ge 0$ for all $x\in(a,b)$, then $f$ is nondecreasing on $[a,b]$ (i.e., $x<y$ implies $f(x)\le f(y)$).
• If $f'(x)\le 0$ for all $x\in(a,b)$, then $f$ is nonincreasing on $[a,b]$.
• If $f'(x)>0$ for all $x\in(a,b)$, then $f$ is strictly increasing on $[a,b]$ (and similarly $f'<0$ implies strictly decreasing).

This is one of the most direct ways analysis turns differential information into global order information, and it is frequently used to prove injectivity and existence of inverses .

Proof sketch: Fix $x<y$ in $[a,b]$. By the mean value theorem there exists $c\in(x,y)$ such that $f(y)-f(x)=f'(c)(y-x).$ If $f'(c)\ge 0$ and $y-x>0$, then $f(y)-f(x)\ge 0$, so $f(x)\le f(y)$. The strict case is identical.