First Derivative Test

Definition. Let f\colon D\to\mathbb{R} be a function. Then a “Local Minima” is a point c\in D such that f(c)\leq f(x) for any x in some open interval containing c.

Likewise a “local maxima” is a point c\in D which has f(c)\geq f(x) for every x in some open interval containing c.

First Derivative Test. If f has a local maxima (or minima) at an interior point c of its domain, and if f'(x) is defined on c, then f'(c)=0.

Proof. We will consider the case for a local minima (the reasoning is very similar for a local maxima). If x>c, then x-c>0. We see that
(1)\displaystyle\frac{f(x)-f(c)}{x-c}\geq0
On the other hand, if x<c, then x-c<0. We see that
(2)\displaystyle\frac{f(x)-f(c)}{x-c}\leq0
Thus by the squeeze theorem we see that
(3)\displaystyle 0\leq\lim_{x\to c}\frac{f(x)-f(c)}{x-c}\leq0
implies f'(c)=0.

Remark. Observe the contrapositive of this theorem states: if f'(c)\not=0, then c cannot be a local maxima or local minima for f.

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About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
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2 Responses to First Derivative Test

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

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

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