Question

判断函数f(x)==SQRT(3)*x+SQRT(1-x^2) 的单调性,过程也要(定义域：（0,1））

Answer 1

此题要利用三角形函数性质,不知道你学了没有

设

x=sint,sint∈(0,1).t∈(0,π/2)

f(x)=√3x+√(1-x^2)

=√3sint+cost

=2sin(t+π/6)

t∈(0,π/2)

t+π/6∈(π/6,2π/3)

2sin(t+π/6)在(π/6,π/2】增在(π/2,2π/3)减

即f(x)在(0,π/3】增,在(π/3,π/2)减

x=sint

增区间是(0,√3/2]

减区间是(√3/2,1)

f(x)=√3x+√(1-x^2)

图像