设函数f(x)=sinx+cosx和g(x)=2sinxcosx试求F(x)=f(x)+ag(x)在[0,0.5π]上的最小值h(a)

题目

设函数f(x)=sinx+cosx和g(x)=2sinxcosx试求F(x)=f(x)+ag(x)在[0,0.5π]上的最小值h(a)

答案

令sinx+cosx=2sin(x+π/4)=t∵0≤x≤π/2,π/4≤x+π/4≤3π/4,∴-√2/2≤sin(x+π/4)≤1即-√2≤t≤2(sinx+cosx)^2=1+2sinxcosx=t^22sinxcosx=t^2-1F(x)=t+a(t^2-1)=at^2+t-a,-√2≤t≤2讨论a取最值当0＜a＜√2/2时...