<prompt> Acha z iwe nambari changamano kama vile z 1. Pata thamani ya juu ya1 z 1 - z z2.  <response>: Acha z x yi, ambapo x na y ni nambari halisi. Tangu z 1, x2 y2 1. Kishaanzaalign1 z 1 - z z2 1 x yi 1 - x - yi x2 2xyi - y2 1 x yi 1 - x x2 - 1 x2 -y 2xyi 1 x yi -x 2x2 -y 2xyi sqrt1 x2 y2 sqrt-x 2x22 -y 2xy2 sqrt1 x2 y2 sqrt-x 2x22 y2 1 - 2x2 sqrt1 x2 1 - x2 sqrt-x 2x22 1 - x2 1 - 2x2 sqrt2 2x sqrt1 - 4x 4x2 sqrt2 2x 1 - 2x.endalignHebu u sqrt2 2x. Kisha u2 2 2x, hivyosqrt2 2x 1 - 2x u 3 - u2.Tangu -1 le x le 1, 0 le u le 2.Ikiwa 0 le u le sqrt3, basiu 3 - u2 u 3 - u2 frac134 - kushoto u - frac12 kulia2 le frac134.Usawa hutokea wakati u frac12, au x -frac78.Ikiwa sqrt3 le u le 2, basiu u2 - 3 kushoto u frac12 kulia2 - frac134 le kushoto 2 frac11 2 kulia2 - frac134 3 frac134.Kwa hivyo, thamani ya juu zaidi ni frac134. 