Question

实数 A=3-根号2-m/1+根号m-1,1≤m≤2则A的取值范围

Answer 1

令m-1=n

则原式等于

(3-根号(1-n))/(1+根号(n)),0≤t≤1

令n=sin²θ

则原式等于

(3-cosθ)/(1+sinθ),0≤θ≤π/2

由万能公式：

设tan(θ/2)=t,0≤t≤1

sinθ=2t/(1+t²)

cosθ=(1-t²)/(1+t²)

原式=(4t²+2)/(t²+2t+1)=4-(8t+2)/(t²+2t+1)

令f(t)=4-(8t+2)/(t²+2t+1)

f(t)在[0,1/2]单调减,在[1/2,1]单调增

f(0)=2,f(1)=3/2,f(1/2)=4/3

∴取值范围为[4/3,2]