<prompt> Acha a,b, c, d ziwe nambari halisi chanya. Pata thamani ya chini ya1 a1 b1 c1 d kushoto frac1a frac1b frac1c frac1d kulia.  <response>: Na AM-GM,1 a frac13 frac13 frac13 a ge 4 sqrt4frac13 3 cdot a 4 sqrt4fraca27.Vile vile,anzaalign1 b ge 4 sqrt4fracb27, 1 c ge 4 sqrt4fracc27, 1 d ge 4 sqrt4fracd27.mwishoalignPia na AM-GM,frac1a frac1b frac1c frac1d ge 4 sqrt4frac1 abcd.Tukizidisha ukosefu huu wote wa usawa, tunapataanzaalign1 a1 b1 c1 d kushoto frac1a frac1b frac1c frac1d kulia ge 4 sqrt4fraca27 cdot 4 sqrt4fracb27 cdot 4 sqrt4fracc27 cdot 4 sqrt4fracd27 cdot 4 sqrt4frac1abcd frac102427.endalignUsawa hutokea wakati a b c d frac13, hivyo thamani ya chini ni frac102427. 