Differentiability Let . For , the derivative of at is . If this exists, then is differentiable at . If exists for all , then is differentiable on .
If is differentiable at , then it is continuous at .
Sequential Criterion differentiable at if and only if , implies converges
has a relative extrema at if where implies is an upper or lower bound for
If exists and is a relative extremum, then
is a two-sided limit if for every , the set and
Interior Extremum Theorem Let be differentiable on . If has max or min value at , then .
Darboux Theorem If differentiable on and , then there is where .
Rolle’s Theorem Let differentiable on . If , then there is a point where .
Mean Value Theorem If is continuous on and differentiable on , then there exists such that