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 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}
 when 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'(c)=0}
 or when 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'(c)}
 is 
 undefined.

Exercises: Find critical numbers of

1) 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)=x^2+2x}

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

2) 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)=\sqrt{x}}

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

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