<prompt> Acha a,b, na c ziwe nambari halisi chanya. Pata thamani ya chini yafraca b ca b2 a b 4c2abc.  <response>: Na AM-GM,a b ge 2 sqrtab,hivyo a b2 ge 4ab.Pia na AM-GM,a 2c b 2c ge 2 sqrta 2cb 2c,hivyo a b 4c2 ge 4a 2cb 2c.Kwa hivyo,anzaaligna b2 a b 4c2 ge 4ab 4a 2cb 2c 8ab 8ac 8bc 16c2 8ab ac bc 2c2.mwishoalignNa AM-GM,anzaalignab ac bc 2c2 fracab2 fracab2 ac bc 2c2 ge 5 sqrt5fracab2 cdot fracab2 cdot ac cdot bc cdot 2c2 5 sqrt5fraca3 b3 c42.mwishoalignPia na AM-GM,anzaaligna b c fraca2 fraca2 fracb2 fracb2 c ge 5 sqrt5fraca2 cdot fraca2 cdot fracb2 cdot fracb2 cdot c 5 sqrt5fraca2 b2 c16.mwishoalignKwa hiyo,anzaalignfraca b ca b2 a b 4c2abc ge 8 cdot frac5 sqrt5 fraca2 b2 c16 cdot 5 sqrt5fraca3 b3 c42abc 100.endalignUsawa hutokea wakati a b 2 na c 1, hivyo thamani ya chini ni 100. 