Manitori

1. Mean value theoremsTheorem 1.1. ($\color{red}{\textrm{Rolle's Theorem}}$) Let $f$ be a function that satisfies the following conditions:$f$ is ${\color{Cyan}{\textrm{continuous}}}$ on the ${\color{orange}{\textrm{closed}}}$ interval $[a,b]$.$f$ is ${\color{Cyan}{\textrm{differentiable}}}$ on the ${\color{orange}{\textrm{open}}}$ interval $(a,b)$.$f(a) = f(b)$Then there exists $\zeta\in(a,b)$ such that $f'(\zeta) = 0$.Theorem 1.2. ($\color{red}{\textrm{The Mean Value Theorem}}$) Let $f$ be a f ...