Toán Lớp 10: Tìm GTLN hoặc GTNN của hàm số y = 5sinx – 12cosx + 15
Toán Lớp 10: Tìm GTLN hoặc GTNN của hàm số y = 5sinx – 12cosx + 15
By Ngọc Quý
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~rai~\(y=5\sin x-12\cos x+15\\\quad=13\left(\dfrac{5}{13}\sin x-\dfrac{12}{13}\cos x\right)+15\\\quad=13\sin(x-\alpha)+15\\\left(\text{với }\alpha=\arccos\dfrac{5}{13}=\arcsin\dfrac{12}{13}\right)\\\text{Ta có:}1\le\sin(x-\alpha)\le 1\\\Leftrightarrow -13\le13\sin(x-\alpha)\le 13\\\Leftrightarrow 2\le13\sin(x-\alpha)\le 28\\\Leftrightarrow 2\le y\le 28.\\+)Min_y=2\Leftrightarrow \sin(x-\alpha)=-1\\\Leftrightarrow x-\alpha=-\dfrac{\pi}{2}+k2\pi\\\Leftrightarrow x=\alpha-\dfrac{\pi}{2}+k2\pi.(k\in\mathbb{Z})\\+)Max_y=28\Leftrightarrow \sin(x-\alpha)=1\\\Leftrightarrow x-\alpha=\dfrac{\pi}{2}+k2\pi\\\Leftrightarrow x=\alpha+\dfrac{\pi}{2}+k2\pi.(k\in\mathbb{Z})\\\text{Vậy Min}_y=2\text{ khi x=}\alpha-\dfrac{\pi}{2}+k2\pi;\\\text{Max}_y=28\text{ khi x=}\alpha+\dfrac{\pi}{2}+k2\pi.(k\in\mathbb{Z})\\\left(\text{với }\alpha=\arccos\dfrac{5}{13}=\arcsin\dfrac{12}{13}\right)\)