Question

f(x)=(sinx+cosx)的平方+2cos平方x-m在x属于R上有零点,则m的取值范围为

Answer 1

f(x)=(sinx+cosx)²+2cos²x-m

=sin²x+2sinxcosx+cos²x+cos(2x)+1-m

=sin(2x)+cos(2x)+2-m

=√2sin(2x+π/4)+2-m

令f(x)=0

√2sin(2x+π/4)=m-2

x∈R,-1≤sin(2x+π/4)≤1 -√2≤√2sin(2x+π/4)≤√2

要方程有解,-√2≤m-2≤√2

2-√2≤m≤2+√2

m的取值范围为[2-√2,2+√2]