Summary of the mean value theorem

Rolle's theorem. If $f$ is continuous on $[a,b]$, differentiable on $]a,b[$, and $f(a)=f(b)$, then $f'$ vanishes somewhere in $]a,b[$.

Mean value theorem (Lagrange). If $f$ is continuous on $[a,b]$ and differentiable on $]a,b[$, then there exists $c\in]a,b[$ with $f(b)-f(a)=f'(c)(b-a)$.

Cauchy's extended MVT. Ratio form $\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{g'(c)}$ when $g'\ne0$.

Consequences.
- $f'=0\Rightarrow f$ constant.
- $f'\ge0\Rightarrow f$ nondecreasing; $f'>0\Rightarrow$ strictly increasing.
- Bounded derivative $\Rightarrow$ Lipschitz.
- Convexity $\iff f'$ nondecreasing.

Inequalities. $\sin x<x$, $\ln x<x-1$, $e^x>1+x$, $\arctan t>t/(1+t^2)$.

Uniqueness of roots. Rolle + $f'>0$ gives at most one root.

Darboux theorem. $f'$ has the intermediate value property.

Conclusion. The MVT is one of the most powerful tools in single-variable calculus, connecting derivatives to function behavior.