MATH 251 Lecture 16

« previous | Friday, February 24, 2012 | next »

Critical points 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(x,y) = xy\sin{\left(\pi(x-y)\right)}} bounded by 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 y= \pm \frac{x}{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 x^2-y^2=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=0} .

${\begin{aligned}{\frac {\partial f}{\partial x}}&=y\sin {\left(\pi (x-y)\right)}+\pi xy\cos {\left(\pi (x-y)\right)}\\{\frac {\partial f}{\partial x}}&=x\sin {\left(\pi (x-y)\right)}-\pi xy\cos {\left(\pi (x-y)\right)}\\{\mbox{both true}}&{\begin{cases}f_{x}=0\rightarrow y\left(\sin {\left(\pi (x-y)\right)}+\pi x\cos {\left(\pi (x-y)\right)}\right)=0\\f_{y}=0\rightarrow x\left(\sin {\left(\pi (x-y)\right)}-\pi y\cos {\left(\pi (x-y)\right)}\right)=0\\\end{cases}}\end{aligned}}$

Therefore either $y=0$ or 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 \sin{\left(\pi(x-y) \right)} + \pi x \cos{\left( \pi(x-y) \right)} = 0} 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=0} or 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 \sin{\left( \pi(x-y) \right)} - \pi y \cos{\left( \pi(x-y) \right)}}

(0,0) is a critical point.

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,\, y\ne0} is not in the domain.

...