<prompt> Acha x na y ziwe nambari halisi chanya kama hizofrac1x 2 frac1y 2 frac13.Tafuta thamani ya chini zaidi ya x 2y.  <response>: Na usawa wa Cauchy-Schwarz,x 2 2y 2 kushoto frac1x 2 frac1y 2 kulia ge 1 sqrt 22.Kishax 2 2y 4 ge 3 1 sqrt22 9 6 sqrt2,hivyo x 2y ge 3 6 sqrt2 .Usawa hutokea wakati x 22 2y 22, au x 2 y 2 sqrt2. Inabadilisha hadi frac1x 2 frac1y 2 frac13, tunapatafrac1y 2 sqrt2 frac1y 2 frac13.Kutatua, tunapata y frac2 3 sqrt22. Kisha x 1 3 sqrt2.Kwa hivyo, thamani ya chini tunayotafuta ni 3 6 sqrt2. 