Question

高中数学问题求解

求f（x）等于x➕跟号下1-x的值域

Answer 1

f(x)=x+√(1-x)
1-x≥0
x≤1
定义域 =(-∞, 1]
f(x)=x+√(1-x)
f'(x)= 1 - 1/[2√(1-x)]
f'(x) =0
1 - 1/[2√(1-x)] =0
2√(1-x) -1 =0
√(1-x) =1/2
1-x = 1/4
x=3/4
f'(x)| x=3/4+ <0 , f'(x)| x=3/4- >0
x=3/4 (max)
max f(x)
= f(3/4)
=3/4+√(1-3/4)
=3/4 +1/2
=5/4
lim(x->-∞) x+√(1-x) ->-∞
值域 =(-∞, 5/4]