设函数f(x)=3sin (ωx -π/6)(ω>0)和g(x)=2cos (2x+φ)+1的图像的对称轴完全相同,若x∈[0,π/2],求f(x)的取值范围
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![设函数f(x)=3sin (ωx -π/6)(ω>0)和g(x)=2cos (2x+φ)+1的图像的对称轴完全相同,若x∈[0,π/2],求f(x)的取值范围](/uploads/image/z/4323416-32-6.jpg?t=%E8%AE%BE%E5%87%BD%E6%95%B0f%28x%29%3D3sin+%28%CF%89x+-%CF%80%2F6%29%28%CF%89%3E0%29%E5%92%8Cg%28x%29%3D2cos+%282x%2B%CF%86%29%2B1%E7%9A%84%E5%9B%BE%E5%83%8F%E7%9A%84%E5%AF%B9%E7%A7%B0%E8%BD%B4%E5%AE%8C%E5%85%A8%E7%9B%B8%E5%90%8C%2C%E8%8B%A5x%E2%88%88%5B0%2C%CF%80%2F2%5D%2C%E6%B1%82f%28x%29%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
设函数f(x)=3sin (ωx -π/6)(ω>0)和g(x)=2cos (2x+φ)+1的图像的对称轴完全相同,若x∈[0,π/2],求f(x)的取值范围
设函数f(x)=3sin (ωx -π/6)(ω>0)和g(x)=2cos (2x+φ)+1的图像的对称轴完全相同,若x∈[0,π/2],
求f(x)的取值范围
设函数f(x)=3sin (ωx -π/6)(ω>0)和g(x)=2cos (2x+φ)+1的图像的对称轴完全相同,若x∈[0,π/2],求f(x)的取值范围
依题意ω=2,
∴f(x)=3sin(2x-π/6),
2x-π/6∈[-π/6,5π/6],
∴f(x)∈[-3/2,3],为所求.
∵f(x)和g(x)的对称轴完全相同,
∴二者的周期相同,即ω=2,f(x)=3sin(2x-6(π)).
∵x∈[0,2(π)],∴2x-6(π)∈[-6(π),6(5π)],sin(2x-6(π))∈[-2(1),1],
∴f(x)∈[-2(3),3].
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