MATH 251 Lecture 16

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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} .

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 \begin{align} \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{align}}

Therefore either 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=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.

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