<prompt> Tafuta nambari kubwa kabisa chanya n kama hiyosinn x cosn x ge frac1nkwa nambari zote halisi x.  <response>: Kuweka x pi, tunapata-1n ge frac1n,kwa hivyo n lazima iwe sawa. Acha n 2m.Kuweka x fracpi4, tunapatakushoto frac1sqrt2 kulia2m kushoto frac1sqrt2 kulia2m ge frac12m.Hii hurahisishafrac12m - 1 ge frac12m,hivyo 2m - 2 le m. Tunaona kwamba m 4 ni suluhu, na chaguo la kukokotoa 2m - 2 hukua haraka kuliko m, hivyo m 4 ndio thamani kubwa zaidi iwezekanayo ya m.Ni lazima basi kuthibitisha hilosin8 x cos8 x ge frac18kwa nambari zote halisi x.Na QM-AM,sqrtfracsin8 x cos8 x2 ge fracsin4 x cos4 x2,hivyosin8 x cos8 x ge fracsin4 x cos4 x22.Tena kwa QM-AM,sqrtfracsin4 x cos4 x2 ge fracsin2 x cos2 x2 frac1 2,hivyosin4 x cos4 x ge frac12.Kwa hiyo,sin8 x cos8 x ge frac1222 frac18.Tunahitimisha kwamba nambari kubwa zaidi chanya kama hiyo n ni 8. 