aec001…: “As for the loud book”

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Find the absolute maximum and minimum of $F(x,y) = x^2+2x+y^2+2y$ constrained to the circle $x^2+y^2=4$ . The critical points on this set are (writing them in the order of increasing $x$ -values):

Closed sets contain their boundry points.

$\begin{align*} \vec{c}_1 &= \vector{\answer{-\sqrt{2}},\answer{-\sqrt{2}}}\\ \vec{c}_2 &= \vector{\answer{\sqrt{2}},\answer{\sqrt{2}}} \end{align*}$

The absolute maximum of $F$ when constrained to the closed set is: $F\left (\answer {\sqrt {2}},\answer {\sqrt {2}}\right ) = \answer {4+4\sqrt {2}}$ The absolute minimum of $F$ when constrained to the closed set is: $F\left (\answer {-\sqrt {2}},\answer {-\sqrt {2}}\right ) = \answer {4-4\sqrt {2}}$