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 
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2}
 in the interval, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2>x_1}
 implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_2)<f(x_1)}

Test for Increasing and Decreasing Functions

 Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}
 be differentiable on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
.
 1. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)>0}
 for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
 in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}
 is increasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
.
 2. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)<0}
 for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
 in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}
 is decreasing on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
.
 3. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Exercises: Find critical numbers of

1) $f(x)=x^{2}+2x$

Solution:
$f'(x)=2x+2=2(x+1)=0$
So, $x=-1$ is critical number

2) $f(x)={\sqrt {x}}$

Solution:
$f(x)=x^{\frac {1}{2}}$
So, $f'(x)={\frac {1}{2}}x^{\frac {-1}{2}}={\frac {1}{2{\sqrt {x}}}}$
In this case, we have critical number when $f'(x)$ is undefined, which is when ${\sqrt {x}}=0$. So critical number is $x=0$

Increasing and Decreasing Test

 1. Find the derivative of $f$.
 2. Locate the critical numbers of $f$ and use these numbers to determine test intervals.
 That is, find all $f$ for which $f'(x)=0$ or $f'(x)$ is undefined.
 3. Determine the sign of $f'(x)$ at one test value in each of the intervals.
 4. Use the test for increasing and decreasing functions to decide 
 whether $f$ is increasing or decreasing on each interval.

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