We want to find the maximum and minimum values of f(x, y) = 10x2 + 11y on the disk D: x2 + y2 < 1.

Now focus on the boundary of D, and solve for y2. Restricting f(x,y) to this boundary, we can express f(x,y) as a function of a single variable x. What is this function and its closed interval domain?

If you're just focusing on the boundary, it's the circle [tex]x^2+y^2=1[/tex]. Parameterize it by taking [tex]x=\cos t[/tex] and [tex]y=\sin t[/tex], with [tex]0\le t\le2\pi[/tex]. Then [tex]f(x,y)[/tex] reduces to a univariate function [tex]g(t)[/tex]:

[tex]f(x,y)=10x^2+11y\iff g(t)=f(\cos t,\sin t)=10\cos^2t+11\sin t[/tex]

Then you can find any extrema on the boundary by checking the critical points of [tex]g[/tex].

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We want to find the maximum and minimum values of f(x, y) = 10x2 + 11y on the disk D: x2 + y2 < 1. Now focus on the boundary of D, and solve for y2. Restricting f(x,y) to this boundary, we can express f(x,y) as a function of a single variable x. What is this function and its closed interval domain?

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