<prompt> Kwa 0 le x le 40 na 0 le y le 50, pata thamani ya chini zaidi yasqrtx2 400 sqrty2 900 sqrtx2 y2 - 80x - 100y 4100.  <response>: Kukamilisha mraba katika x na y, usemi unakuwasqrtx2 400 sqrty2 900 sqrtx - 402 y - 502 sqrtx2 400 sqrty2 900 sqrt40 - x2 50 - y2.Na QM-AM,anzaalignsqrtfracx2 4002 ge fracx 202, sqrtfracy2 9002 ge fracy 302, sqrtfrac40 - x2 50 - y22 ge frac40 - x 50 - y2,mwishoalignhivyoanzaalignsqrtx2 400 sqrty2 900 sqrt40 - x2 50 - y2 ge sqrt2 cdot fracx 202 sqrt2 cdot fracy 302 sqrt2 cdot frac 40 - x 50 -y2 70 sqrt2.endalignUsawa hutokea wakati x 20 na y 30, hivyo thamani ya chini ni 70 sqrt2. 