Question

若AB=2,AC=√2BC,求S△ABC的最大值.

Answer 1

cosA=(AB^2+AC^2-BC^2)/2AB*AC =(4+BC^2)/(4√2BC),sinA=√(1-cos^2A)=√[1-((4+BC^2)/(4√2BC))^2] =√(-BC^4+24BC-16)/(4√2BC),S-△ABC=1/2*AB*AC*sinA =1/4*√(-BC^4+24BC-16) =1/4*√[-(BC^2-12)^2+128].要使S-△ABC有最大值,则BC^2-12=0,BC^2=12,BC=2√3时,S-△ABC有最大值,S-△ABC最大值=1/4*√128=2√3.