Question $4(35\%)$

Let $f(x,y)=2x^3-6xy+3Ry+5,$ on the domain $D$ defined by $0x2R$ and $0yR.$

$($a$)$ Find the critical point$($s$)$ of the function $f$ in the interior of the domain $D.$

$(5$ marks$)$

$($b$)$ Find the extremum values of $f$ on the lower $x$ boundary of $D.$

$(5$ marks$)$

$($c$)$ Find the extremum values of $f$ on the upper $x$ boundary of $D.$

$(5$ marks$)$

$($d$)$ Find the extremum values of $f$ on the lower $y$ boundary of $D.$

$(5$ marks$)$

$($e$)$ Find the extremum values of $f$ on the upper $y$ boundary of $D.$

$($f$)$ Hence, find the absolute extrema of the function $f$ on the domain $D.$