MATH 251 Lecture 37

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Review Session

Lagrange Multipliers

$f(x,y,z)=xyz$, constraint: $g(x)=x^{2}+2y^{2}+3z^{2}=6$

Find extrema of $f$ given $g=6$.

${\begin{aligned}{\vec {\nabla }}f&=\lambda {\vec {\nabla }}g\\yz&=\lambda 2x\\xz&=\lambda 4y\\xy&=\lambda 6z\\x^{2}+2y^{2}+3z^{2}&=6\end{aligned}}$

Solving the system of equations gives

${\begin{aligned}x&=\pm {\sqrt {2}}\\y&=\pm 1\\z&=\pm {\sqrt {\tfrac {2}{3}}}\end{aligned}}$

for a total of 8 possible points.

(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 \lambda} is meaningful only 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 Q(x,y) = ax^2+2bxy+cy^2} is a quadric surface 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 g(x,y) = x^2+y^2=1} is the unit circle)

Finding Extrema in a region

Find the extrema 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) = 2x^2+x+y^2-2} within 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 D = \{ x^2+y^2=4 \}} .

1. look interior: 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 \vec\nabla f = 0}
2. look on boundary: 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) = x^2+x+2} (substitute 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=4} )

Solving 1 gives us 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 (-\tfrac{1}{4}, 0)} easily.

Solve for 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 \begin{align} x&=2\cos{\theta}\\ y&=2\sin{\theta}\\ f(\theta) &= 8\cos^2{\theta} + 2\cos{\theta} + 4\sin^2{\theta} - 2 \\ f'(\theta) &= (2\sin{\theta})(4\cos{\theta}+1) = 0 \\ \theta &= 0, \pi \\ \cos{\theta} &= -\tfrac{1}{4} \end{align}}

We now have 5 points to check: (-2,0), (2,0), (-1/2, ±sqrt(15)/2), and (-1/4,0)