Consider the function $f(x,y)=x^2-6yx-2x+9y^2+6y+5.We$ claim that $f$ has infinitely many critical points.

$(a)In$ fact, the critical points $off$ lie $on$ a line $x=ky+b.$ Find the equation $of$ this line.

$k=$

$b=$

$(b)$ Confirm that the function can $be$ rewritten $in$ the form $f(x,y)=(x-ky-b{)}^2+4,$ where $k$ and $b$ are the values found above. What can $be$ said about the nature $of$ the critical

points $off(i.e.$ the points $on$ the line$)?$

The critical points are all loca$\frac{l}{a}$bsolute minima.

The critical points are all saddle points.

$No$ information can $be$ deduced.

The critical points are all loca$\frac{l}{a}$bsolute maxima.

$(c)We$ say that a function has a translational symmetry $if$ there $is$ a vector $(a,b)so$ that $f(x+at,y+bt)=f(x,y)$ for all $tx,yf$ has infinitely many critical points, because

$f_x(x,y)is$ a multiple $of{f}_y(x,y).$

$It$ has translational symmetry.

$Itis$ a quadratic surface.

$(d)(i)$ Compute the value $Dof$ the determinant $in$ the second derivative test for these critical points.

$D=$

$(d)(ii)Is$ this consistent with the conclusion $in(c)?$

$No,$ because $D$ would have $tobe$ positive.

$No,$ because $D$ would have $tobe$ negative.

Yes, because $D=0$ gives $no$ further information about the character $of$ the critical points.

Yes, because there are infinitely many critical points.