23.在△ABC中,sinAcosB+cosAsinB=√3/2,角C为钝角,

23。1）∵sinAcosB+cosAsinB=√3/2,∴sin（A+B）=√3/2∵sin(A+B)=sin(π-C)=sinC∴sinC=√3/2 ∵角C为钝角∴C=2π/32）∵c=2√3,根据余弦定理：c²=a²+b²-2abcosC∴a²+b²+ab=12 (1º)∵ a+b=5,∴a²+b²+2ab=25 (2º)(2º)-(1º):ab=13SΔABC=1/2* absinC=13√3/4 24。f(x)=-√2sinxcosx+√2cos^2(x)+√2/2=-√2/2*sin2x+√2/2(1+cos2x)+√2/2=√2/2cos2x-√2/2*sin2x +√2=cos(2x+π/4)+√21）f(x)最小正周期T=2π/2=π2) ∵x∈[0,π],∴2x∈[0,2π] ∴ 2x+π/4 ∈ [π/4,9π/4]2x+π/4 从π,增到2π,f(x)由最小增到最大此时,x∈[3π/8,7π/8]∴f(x)在[0,π]上的单调递增区间是[3π/8,7π/8]3) ∵x∈[0,π/2] ∴2x∈[0,π] 2x+π/4 ∈[π/4,5π/4]∴ 2x+π/4=π/4,即x=0时,f(x)max=1+√22x+π/4=π,即x=3π/8时,f(x)max=-1+√2