<prompt> Tafuta kiwango cha juu kabisa cha y-coordinate cha nukta kwenye grafu ya r sin 2 theta.  <response>: Kwa r sin 2 theta,anzaaligny r dhambi theta dhambi 2 theta sin theta 2 sin2 theta cos theta 2 1 - cos2 theta cos theta.mwishoalignAcha k cos theta. Kisha y 2 1 - k2 k, nay2 4k2 1 - k22 4k2 1 - k2 1 - k2.Na AM-GM,2k2 1 - k21 - k2 le left frac2k2 1 - k2 1 - k2 3 kulia3 frac827,hivyoy2 le frac1627.Kwa hiyo,y le sqrtfrac1627 frac4 sqrt39.Tunapata y frac4 sqrt39 wakati k2 cos2 theta frac13, kwa hivyo hiki ndicho kiwango cha juu zaidi cha kuratibu y-.asyukubwa wa kitengo cm 3;jozi moo t halisi halisi r dhambi 2 t; kurudi rcost, rsint;njia foo moo0;t halisi;kwa t 0; t 2pi 0.01; t t 0.01 foo foo--moot;chorafoo,nyekundu;chora-1,0--1,0;chora0,-1--0,1;chora-1,4sqrt39--1,4sqrt39,bluu;lebor sin 2 theta, 1.2,0.6, nyekundu;leboy frac4 sqrt39, -1, 4sqrt39, W, bluu;asy 