Definition. Let be a function. Then a “Local Minima” is a point such that for any in some open interval containing .
Likewise a “local maxima” is a point which has for every in some open interval containing .
First Derivative Test. If has a local maxima (or minima) at an interior point of its domain, and if is defined on , then .
Proof. We will consider the case for a local minima (the reasoning is very similar for a local maxima). If , then . We see that
On the other hand, if , then . We see that
Thus by the squeeze theorem we see that
implies . ∎
Remark. Observe the contrapositive of this theorem states: if , then cannot be a local maxima or local minima for .