<prompt> Acha a, b, c, d ziwe nambari halisi chanya ili a b c d 1. Pata thamani ya chini zaidi yafrac1a frac1b frac4c frac16d.  <response>: Na Cauchy-Schwarz,a b c d kushoto frac1a frac1b frac4c frac16d kulia ge 1 1 2 42 64. Usawa hutokea wakati a b fracc2 fracd4 na a b c d 1. Tunaweza kutatua ili kupata a frac18, b frac18, c frac14 , na d frac12, kwa hivyo thamani ya chini ni 64. 