Question

Gọi M là giá trị lớn nhất của biểu thức \(S=\sin x+\sin y+\sin\left(3x+y\right)-2\sin\left(2x+y\right).\cos x\) , \(\forall x\in\left(0,2\pi\right),\forall y\in\left(0,2\pi\right)\) . Biết \(M=\dfrac{a\sqrt{b}}{c}\) (Với a,b,c \(\in Z^+,\dfrac{a}{c}\) là phân số tối giản, b < 12). Tính \(P=a+b-c\)

Answer 1

\(S=sinx+siny+sin\left(3x+y\right)-sin\left(3x+y\right)-sin\left(x+y\right)\)

\(=sinx+siny-sin\left(x+y\right)\)

\(S^2=\left(sinx+siny-sin\left(x+y\right)\right)^2\le3\left(sin^2x+sin^2y+sin^2\left(x+y\right)\right)\)

\(S^2\le3\left(1-\dfrac{1}{2}\left(cos2x+cos2y\right)+sin^2\left(x+y\right)\right)\)

\(S^2\le3\left[1-cos\left(x+y\right)cos\left(x-y\right)+1-cos^2\left(x-y\right)\right]\)

\(S^2\le3\left[2+\dfrac{1}{4}cos^2\left(x+y\right)-\left[cos\left(x-y\right)-\dfrac{1}{2}cos\left(x+y\right)\right]^2\right]\le3\left[2+\dfrac{1}{4}cos^2\left(x+y\right)\right]\)

\(S^2\le3\left(2+\dfrac{1}{4}\right)=\dfrac{27}{4}\)

\(\Rightarrow S\le\dfrac{3\sqrt{3}}{2}\)

\(\Rightarrow\left\{{}\begin{matrix}a=3\\b=3\\c=2\end{matrix}\right.\)