**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

(1)

On the other hand, if , then . We see that

(2)

Thus by the squeeze theorem we see that

(3)

implies . ∎

*Remark*. Observe the contrapositive of this theorem states: if , then cannot be a local maxima or local minima for .

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Pingback: [Index] Calculus of a Single Variable | My Math Blog

Pingback: [Index] Calculus in a Single Variable | My Math Blog