<prompt> Acha x, y, na z ziwe nambari halisi chanya kama hizofrac1x4 frac1y4 frac1z4 1.Tafuta thamani ya chini zaidi yafracx4 y4 x4 z4 y4 z4x3 y2 z3.  <response>: Tunaweza kuandikaanzaalignfracx4 y4 x4 z4 y4 z4x3 y2 z3 fracxy2 zx4 y4 x4 z4 y4 z4x4 y4 z4 xy2 z cdot kushoto frac1x4 frac1y4 frac1z4 kulia xy2 z.endalignSasa, na AM-GM,anzaalignfrac1x4 frac1y4 frac1z4 frac1x4 frac1 2y4 frac12y4 frac1z4 ge 4 sqrt4frac1x4 cdot frac12y4 cdot frac12y4 cdot frac1 z4 frac2 sqrt2xy2 z,endalignhivyo xy2 z ge 2 sqrt2.Usawa hutokea wakati x4 2y4 z4; pamoja na sharti frac1x4 frac1y4 frac1z4 1, tunaweza kutatua ili kupata x sqrt2, y sqrt42, na z sqrt2, hivyo thamani ya chini ni 2 sqrt2. 