theorems

Fermat’s Theorem says that local extrema of a differentiable function occur at points where the first derivative is zero, meaning the tangent line is horizontal.

The Cauchy theorem states that, under certain conditions of continuity and differentiability for two functions \( f(x) \) and \( g(x) \), there exists at least one point in the interval where the ratio of their derivatives equals the ratio of their increments at the endpoints of the interval.

\[
\frac{f’ \left(c \right)}{g’ \left(c \right)} = \frac{f(b) – f(a)}{g(b) – g(a)}
\]