- 三角函数的性质已知函数f(x)=2sin^2(pai/4+x)
- 已知f(x)=2sin^2(pai/4+x)-√3sos2s,x属于
1.求f(x)的最大值和最小值
2.若不等式|f(x)-m|<2在x属于[pai/4,pai/2]上恒成立，求实数m的取值范围.
- 已知f(x)=2sin^2(x+π/4) -√3cos2x, x∈[π/4,π/2]
1.求f(x)的最大值和最小值
f(x)=2sin^2(x+π/4) –√3cos2x,
=1 – cos(2x+π/2) –√3cos2x
=1 +sin2x –√3cos2x
=1 +2 sin(2x –π/3), x∈[π/4,π/2]
∵x∈[π/4,π/2]
∴(2x –π/3)∈[π/2–π/3, π–π/3]
f(x)的最大值是
f(5π/12)= 1 +2sin(π–π/2)=1+2sin(π/2)=1+2=3
f(x)的最小值是
f(π/4)= 1 +2sin(π/2–π/3)=1+2cos(π/3)=1+1=2
2.若不等式|f(x) –m|<2在x∈[π/4,π/2]上恒成立，求实数m的取值范围.
∵|f(x) –m| <2在x∈[π/4,π/2]上恒成立
∴|3 –m| <2且|2 –m| <2
从而1＜m＜4.(在数轴上确定m的范围)