Consequences of Lagrange: increase and decrease

From the hypotheses of Lagrange’s theorem the links between the sign of the derivative and the behaviour of the function follow directly: these are the properties that make the derivative the foremost tool for studying a function.

Property — Fundamental consequences of Lagrange

• If $f'(x)>0$ for every $x\in(a,b)$ $\implies$ $f$ is strictly increasing on $[a,b]$.
• If $f'(x)<0$ for every $x\in(a,b)$ $\implies$ $f$ is strictly decreasing.
• If $f'(x)=0$ for every $x\in(a,b)$ $\implies$ $f$ is constant.

These consequences turn the study of the sign of the derivative into the main tool for understanding where a function increases and where it decreases.

Links

Topics: Calculus theorems
Concepts: Increase and decrease · Derivative · Lagrange’s theorem
Skills: Studying a function
People: Joseph-Louis Lagrange