Question

已知a属于R 求函数y=(a-sinx)(a-cosx)的最小值

Answer 1

∵y=(a-sinx)(a-cosx)=a^2 -a(sinx+cosx) +sinxcosx,

令sinx+cosx=t,t∈[-√2,√2],则sinxcosx=(t^2 -1)/2. 则y=a^2 -at+(t^2 -1)/2=[(t-a)^2 +a^2 -1]/2,

记f(t)=[(t-a)^2 +a^2 -1]/2, t∈[-√2,√2],分类讨论如下,

①当a≥√2时,f(t)在[-√2,√2]单调递减,则ymin=f(√2)=a^2 -√2a +1/2

②当-√2