Question

Find all points where the function has any relative extrema or saddle points and identify the type of relative extremum. f(x,y)= e-(x2 + y2 -by) A. Relative maximum at (0,3) OB. Saddle point at (0,3) O C. Relative maximum at at (0,3) and relative minimum at at (0, -3) OD. No relative extremum or saddle points.

Answer 1

It is asked to find relative extrema or sadd ce points of flory)=e 268) x² + y² function Criftical a fanction f(x,y) is the =0 Note The relative extrema er scoldle point of a fenetén fevry) occer at critical points. g, thue points of and pint where for fg - for classification the critical punt Suppose (26,9o) is a critical point of f@,9) det D = fr&(2,4).fgy(80, 4) – fry (70,90) If oso and fxx (2o, Ge) so the a relatiue minimna m af we use the fellowing test. • 2 there is o, go) If Do and facx (xo, Go) to the there & relative maximum at (70,90 а

Olf 020 then perint (7,5) ej a sadalle point If D=0 no conclusions can be made x 4469 X-20 -2xe x ² g²464) dal · f@r,y) = 0 - (x + y²-68) ->$q-2(-7-9'46) ex-y²rby fax=2 (-axe get yleep 23-axe_2+3 = 2 x - y² +64 {1- 2x² • fy = 2 ( ex² - y² +64 1-24 +6] -X-4 +68 2 dy 2 é - x _y²tby

4 fyy за I e-z'-8°44 Г-ау +6 ] -a) +6 ] 1-364 (-84 +6) {-2} +6)+e -z'g'464 С (-2444 )? 27 • 2,4 - 2 дч 1*-*+64 Я ou faze-x+y+6 -2x ex _q? 64 - ач + 6 7 -ax [-24ts Je-x-y° +44 find critical point equate fx = 0 and fy=0 -21 е-ху° 61 о — Jr -ат е

ex²_9²469 70 2 - x²-y² +69 • fy = fay+6)e x²-y ² +6 y 0 t -2y+6=0 y = 6 ie Nocu eve We have only one critical peint (0,3) hane to identify the getuie ģ the vitical point fxx(0,3) -26 0² 3²76x3 [1-2x0² : -2e +9 Lo O - fyy (0, 3) = &-0² 3²46x3 { (-2+3+6) ² - 2] -209 "det 9

-0?_3?+2x3 fag (0, 3) = rako{"ax3 46') e 2 => D= fax (0,3). fgg (0, 3) – fax (0,3) -2e x-269 - 0² 9 18 He D=468 <o We have res do and fro(0, 3) = -30 .. we have Relatene maximum at (0,3) · The correct option A of select A Relatine waumnum at (0,3 iš