Question

求函数的值域（如图）

求函数的值域（如图）那里是4次方，图不太清晰。

Answer 1

y=(sin(x/2))^4+(cos(x/2))^4=((sin(x/2))^2+(cos(x/2))^2)^2-2*(sin(x/2))^2*(cos(x/2))^2
=1-(2*(sin(x/2)*(cos(x/2))^2/2=1-(sin(2x))^2/2
由sin(2x)∈【-1,1】得 (sin(2x))^2∈【0,1】 (sin(2x))^2/2∈【0,1/2】
所以1-(sin(2x))^2/2∈【0,1/2】即 y∈【0,1/2】函数值域为∈【0,1/2】