Question

已知函数f(x)=sin(2x)-a(sinx+cosx)的最小值为g(a) 求g(a)

Answer 1

f(x)=2sinxcosx-asinx-acosx

令sinx+cosx=t=(√2)sin(x+π/4)

则-√2≤t≤√2

f(x)=t^2-1-at=t^2-at-1=(t-a/2)^2-1-a^2/4

1）当-√2≤a/2≤√2,即-2√2≤a≤2√2时

t=a/2时取到最小值-1-a^2/4

2）当a/2>√2即a>2√2时,t=√2时取到最小值1-√2a

3）当a/2