作业帮如题.
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/19 21:54:18
如题.
如题.
如题.
(1)
x = π/6,f(x) = (1/2)sin(π/3)sinφ + [cos²(π/6)]cosφ - (1/2)cosφ
= (√3/4)sinφ + (3/4)cosφ - (1/2)cosφ
= (1/2)[(√3/2)sinφ + (1/2)cosφ]
= (1/2)[sinφcos(π/6) + cosφsin(π/6)]
= (1/2)sin(φ + π/6) = 1/2
sin(φ + π/6) = 1
φ + π/6 = π/2
φ = π/3
f(x) = (√3/4)sin2x + (1/2)cos²x - 1/4
(2)横坐标为原来的一半,x变为x/2:
g(x) = (√3/4)sinx + (1/2)cos²(x/2) - 1/4
= (√3/4)sinx + (1/4)(cosx + 1) - 1/4
= (√3/4)sinx + (1/4)cosx
= (1/2)[(√3/2)sinx + (1/2)cosx]
= (1/2)sin(x + π/6)
0 ≤ x ≤ π/4,π/6 ≤ x + π/6 ≤ 5π/12 < π/2,g(x)为增函数
最小值g(0) = (1/2)sin(π/6) = 1/4
最大值g(π/4) = (1/2)sin(5π/12) = (1/2)sin(π/4 + π/6)
= (1/2)sin(π/4)cos(π/6) + (1/2)cos(π/4)sin(π/6)
= (1/2)(√2/2)(√3/2) + (1/2)(√2/2)(1/2)
= √2(√3 + 1)/8
cos^2x = 1/2 (1+cos2x)
f(x) = 1/2 sin2x sinPhi + 1/2 cosPhi + 1/2 cos2xcosPhi - 1/2 cosPhi = 1/2 cos(2x-Phi)
f(pi/6) = 1/2, cos(2pi/3 - phi) = 1, 2pi/3 - phi = 0, phi = 2pi/3
f(0) = -1/4
f(pi/4) = sqrt(3)/4
g(x)的最小值为-1/8，最大值sqrt(3)/8