Math 22 Increasing and Decreasing Functions

==Definitions of Increasing and Decreasing Functions.

 A function  is increasing on an interval when, for any two numbers $x_{1}$ and 
 $x_{2}$ in the interval, $x_{2}>x_{1}$ implies $f(x_{2})>f(x_{1})$

 A function  is decreasing on an interval when, for any two numbers $x_{1}$ and 
 $x_{2}$ in the interval, $x_{2}>x_{1}$ implies $f(x_{2})<f(x_{1})$

Test for Increasing and Decreasing Functions

 Let $f(x)$ be differentiable on the interval $(a,b)$.
 1. If $f'(x)>0$ for all $x$ in $(a,b)$, then $f$ is increasing on $(a,b)$.
 2. If $f'(x)<0$ for all $x$ in $(a,b)$, then $f$ is decreasing on $(a,b)$.
 3. If $f'(x)=0$ for all $x$ in $(a,b)$, then $f$ is constant on $(a,b)$.

Critical Numbers and Their Use

 If $f$ is defined at $c$, then $c$ is a critical number of $f$ 
when $f'(c)=0$ or when $f'(c)$ is undefined.

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