<prompt> Acha a, b, c, d, e, f ziwe nambari halisi chanya ili a b c d e f 7. Pata thamani ya chini zaidi yafrac1a frac4b frac9c frac16d frac25e frac 36f.  <response>: Na Cauchy-Schwarz,a b c d e f kushoto frac1a frac4b frac9c frac16 d frac25e frac36f kulia ge 1 2 3 4 5 62 441,hivyofrac1a frac4b frac9c frac16d frac25e frac 36f ge frac4417 63.Usawa hutokea wakati a2 fracb24 fracc29 fracd216 frace225 fracf236 na a b c d e f 7. Kutatua, tunapata a frac13, b frac23, c 1, d frac43, e frac53, na f 2, hivyo thamani ya chini ni 63. 