<prompt> Acha x na y ziwe nambari halisi chanya. Pata thamani ya chini yakushoto x frac1y kulia kushoto x frac1y 2018 kulia kushoto y frac1x kulia kushoto y frac1x 2018 kulia.  <response>: Na QM-AM,sqrtfracx frac1y2 y frac1x22 ge fracx frac1y y frac1x2,hivyokushoto x frac1y kulia2 kushoto y frac1x kulia2 ge frac12 kushoto x frac1y y frac1x kulia2.Kishaanzaalignkushoto x frac1y kulia kushoto x frac1y 2018 kulia kushoto y frac1x kulia kushoto y frac1x 2018 kulia kushoto x frac1y kulia2 kushoto y frac1x kulia2 2018 kushoto x frac1 y kulia 2018 kushoto y frac1x kulia ge frac12 kushoto x frac1y y frac1x kulia2 2018 kushoto x frac 1y y frac1x kulia frac12 u2 2018u,endalignambapo u x frac1y y frac1x.Na AM-GM,u x frac1x y frac1y ge 2 2 4.Kitendaji frac12 u2 2018u inaongezeka kwa u ge 4, hivyofrac12u2 2018u ge frac12 cdot 42 2018 cdot 4 8080.Usawa hutokea wakati x y 1, hivyo thamani ya chini ni 8080. 