Question

已知向量a=(√3sinx/2,cosx/2),b=(cosx/2,cosx/2)令f(x)=向量a乘向量b，求f(x)+f`(x)的值域

如题

Answer 1

f(x) =a.b
= (√3sinx/2,cosx/2).(cosx/2,cosx/2)
=√3sinx/2cosx/2+cosx/2cosx/2
= (√3/2)sinx + (1/2)cosx + 1/2
= sin(x+π/6) +1/2
f'(x) = cos(x+π/6)
f(x) + f'(x) = sin(x+π/6) +1/2 + cos(x+π/6)

consider
g(x)= sin(x+π/6) + cos(x+π/6)
g'(x) = cos(x+π/6) - sin(x+π/6) =0
tan(x+π/6) = 1
x = π/12(max) or 13π/12(min)
max f(x) +f'(x) = sin(π/4) +1/2 + cos(π/4)
= √2 +1/2
min f(x) +f'(x) = sin(5π/4) +1/2 + cos(5π/4)
= -√2 +1/2
域值=[-√2 +1/2,√2 +1/2]