<prompt> Acha a na b ziwe nambari halisi chanya kiasi kwamba a 2b 1. Pata thamani ya chini zaidi yafrac1a frac1b.  <response>: Na Cauchy-Schwarz,a 2b kushoto frac1a frac1b kulia ge 1 sqrt22 3 2 sqrt 2.Ili usawa ufanyike, lazima tuwe na a2 2b2, au a b sqrt2. Kisha b sqrt2 2b 1, aub frac12 sqrt2 frac2 - sqrt22 sqrt22 - sqrt2 frac2 - sqrt22,na a b sqrt2 frac2 sqrt2 - 22 sqrt2 - 1.Kwa hivyo, thamani ya chini ni 3 2 sqrt2. 