$\newenvironment {prompt}{}{} \newcommand {\DeclareMathOperator }[2]{\@OldDeclareMathOperator {##1}{##2}\immediate \write \myfile {\unexpanded {\DeclareMathOperator }{\unexpanded {##1}}{\unexpanded {##2}}}} \newcommand {\amatrix }[1]{\left [\begin {array}{@{}*{##1}{c}|c@{}} } \newcommand {\RR }[0]{\bf R} \newcommand {\adj }[0]{\mathop {\rm adj}} \newcommand {\degrees }[0]{^{\circ }} \newcommand {\Col }[0]{\mathop {\rm Col}} \newcommand {\diag }[0]{\mathop {\rm diag}} \newcommand {\Inn }[0]{\mathop {\rm Inn}} \newcommand {\Nul }[0]{\mathop {\rm Nul}} \newcommand {\rank }[0]{\mathop {\rm rank}} \newcommand {\rk }[0]{\mathop {\rm rk}} \newcommand {\Row }[0]{\mathop {\rm Row}} \newcommand {\proj }[0]{\mathop {\rm proj}} \newcommand {\Span }[0]{\mathop {\rm Span}} \newcommand {\spec }[0]{\mathop {\rm spec}} \newcommand {\tr }[0]{\mathop {\rm tr}} \newcommand {\vol }[0]{\mathop {\rm vol}} \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[2]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{} \newcommand {\AppendGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Append \grfext@Check }{\grfext@Append \grfext@@Add }} \newcommand {\PrependGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Prepend \grfext@Check }{\grfext@Prepend \grfext@@Add }} \newcommand {\RemoveGraphicsExtensions }[1]{\@ifundefined {Gin@extensions}{\def \Gin@extensions {}}{\edef \grfext@tmp {\zap@space ##1 \@empty }\@for \grfext@ext :=\grfext@tmp \do {\def \grfext@next {\let \grfext@tmp \Gin@extensions \@expandtwoargs \@removeelement \grfext@ext \Gin@extensions \Gin@extensions \ifx \grfext@tmp \Gin@extensions \let \grfext@next \relax \fi \grfext@next }\grfext@next }}\grfext@Print \RemoveGraphicsExtensions } \newcommand {\epstopdfsetup }[0]{\setkeys {ETE}} \newcommand {\epstopdfcall }[1]{\ifETE@InsideSetfile \expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi {‘##1}{\Gin@base \Gin@ext }} \newcommand {\epstopdfDeclareGraphicsRule }[4]{\ifx \\##4\\\@PackageError {epstopdf-base}{Conversion command is missing}\@ehc \else \begingroup \@ifundefined {Gin@rule@##1}{}{\@PackageInfo {epstopdf-base}{Redefining graphics rule for ‘##1’}}\endgroup \@namedef {Gin@rule@##1}####1{{##2}{##3}{\epstopdfcall {##4}}}\fi } \newcommand {\mdf@patchamsthm }[0]{\let \mdf@deferred@thm@head \deferred@thm@head \pretocmd {\deferred@thm@head }{\@inlabelfalse }{\mdf@PackageInfo {mdframed detected package amsthm changed the theorem header of amsthm\MessageBreak }}{\mdf@PackageError {mdframed detected package amsthm changed the theorem header of amsthm failed\MessageBreak }}} \newcommand {\GetTitleStringSetup }[0]{\setkeys {gettitlestring}} \newcommand {\GetTitleString }[0]{\ifGTS@expand \expandafter \GetTitleStringExpand \else \expandafter \GetTitleStringNonExpand \fi } \newcommand {\GetTitleStringExpand }[1]{\def \GetTitleStringResult {##1}\begingroup \GTS@DisablePredefinedCmds \GTS@DisableHook \edef \x {\endgroup \noexpand \def \noexpand \GetTitleStringResult {\GetTitleStringResult }}\x } \newcommand {\GetTitleStringNonExpand }[1]{\def \GetTitleStringResult {##1}\global \let \GTS@GlobalString \GetTitleStringResult \begingroup \GTS@RemoveLeft \GTS@RemoveRight \endgroup \let \GetTitleStringResult \GTS@GlobalString } \newcommand {\GTS@DisableHook }[0]{} \newcommand {\label@hook }[0]{} \newcommand {\@currentlabelname }[0]{} \newcommand {\reserved@a }[0]{} \newcommand {\@safe@activestrue }[0]{} \newcommand {\@safe@activesfalse }[0]{} \newcommand {\nameref }[0]{\@ifstar \T@nameref \T@nameref } \newcommand {\Sectionformat }[2]{##1} \newcommand {\vnameref }[1]{\unskip ~\nameref {##1}\@vpageref [\unskip ]{##1}} \newcommand {\ref }[0]{\@ifstar \@refstar \T@ref } \newcommand {\pageref }[0]{\@ifstar \@pagerefstar \T@pageref } \newcommand {\nameref }[0]{\@ifstar \@namerefstar \T@nameref } \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{}$
Compute the maximum and minimum values attained by the quadratic form $$, where $$ subject to the constraint $$, and determine a unit vector where each extremum is attained.

Maximum = 6 is attained at $$
Minimum = -4 is attained at $$

Find the maximum value for the quadratic form corresponding to the matrix $$ and the vector $$ you computed above, subject to $$ and $$.

Maximum = -4 is attained at $$

Find the maximum value for the quadratic form:
$$, subject to $$.

Maximum = $$.

Given the orthogonal diagonalization of $$ below: ,

Find the unit vector $$ at which the quadratic form $$ attains its maximum subject to $$. Then find the unit vector $$ at which $$ attains its maximum subject to both $$ and $$.

$$.