Question

求值域，函数＝根号（x-4)+根号(15-3x) 用三角换元法，跪求

Answer 1

令根号（x-4)=cost
根号(15-3x)=（根号3）sint
则满足3*[根号（x-4)]²+[根号(15-3x)]²=3
cost+(根号3）sint
=2sin(t+π/6)
函数定义域为使cost和sint非负的t值，可取0<=t<=π/2
则π/6<=t+π/6<=2π/3
所以sin(t+π/6)取值范围是[1/2,1]
函数的值域为[1,2]

Answer 2

(x-4)+(5-x)=1
设 x-4=cos²A 5-x=sin²A A∈[0,90°]
得 函数=cosA+√3sinA
=2sin(A+30°)
A+30°∈[30°，120°]
所以 值域为 [1,2]