<prompt> Pembetatu ABC ina angle C 60circ na BC 4. Pointi D ni sehemu ya kati ya BC. Ni thamani gani kubwa inayowezekana ya tanangle BAD?mathrmA fracsqrt36qquadmathrmB fracsqrt33qquadmathrmC fracsqrt32sqrt2qquadmathrmD fracsqrt34sqrt2-3 qquadmathrmE 1  <response>: asyunitsize12mm; jozi C0,0, B4 dir60, A 8,0, D2 dir60; jozi E1,0, F2,0; kuchoraC--B--A--C; choraA--D;choraD--E;choraB--F; nuktaA; nuktaB;kitoneC;kitoneD;kitoneE;kitoneF; leboC,C,SW; leboB,B,N; leboA,A,SE; leboD,D,NW; leboE,E,S; leboF,F,S; lebo60circ,C.1,.1,ENE; lebo2,1dir60,NW; lebo2,3dir60,NW; lebotheta,7,.4; lebo1,.5,0,S; lebo1,1.5,0,S; lebox-2,5,0,S;asyAcha x CA . Kisha tantheta tanangle BAF - angle DAE, na kwa kuwa tanangle BAF frac2sqrt3x-2 na tan angle DAE fracsqrt3x-1, tunayotantheta fracfrac2sqrt3x-2 - fracsqrt3x-11 frac2sqrt 3x-2cdotfracsqrt3x-1 fracxsqrt3x2-3x8Kwa calculus, kuchukua derivative na kuweka sawa na sifuri itatoa thamani ya juu ya tan theta. Vinginevyo, tunaweza kuomba AM-GM:anzaalign fracx2 - 3x 8x kushotox frac8xkulia -3 geq 2sqrtx cdot frac 8x - 3 4sqrt2 - 3 fracxx2 - 3x 8 leq frac14sqrt2-3 fracxsqrt3x2 - 3x 8 tan theta leq fracsqrt34sqrt2-3endalign Kwa hivyo, kiwango cha juu ni fracsqrt34sqrt2-3. 