Tìm tất cả các hàm $f:\mathbb{R}\rightarrow \mathbb{R}$ thỏa mãn:
$\left\{\begin{matrix}f(0)=2016,f(\frac{\pi}{2})=2017 \\ f(x+y)+f(x-y)=2f(x)cosy,\forall x,y \in \mathbb{R} \end{matrix}\right.$
Tìm tất cả các hàm $f:\mathbb{R}\rightarrow \mathbb{R}$ thỏa mãn:
$\left\{\begin{matrix}f(0)=2016,f(\frac{\pi}{2})=2017 \\ f(x+y)+f(x-y)=2f(x)cosy,\forall x,y \in \mathbb{R} \end{matrix}\right.$
$$\mathbf{\text{Every saint has a past, and every sinner has a future}}.$$
Tìm tất cả các hàm $f:\mathbb{R}\rightarrow \mathbb{R}$ thỏa mãn:
$\left\{\begin{matrix}f(0)=2016,f(\frac{\pi}{2})=2017 \\ f(x+y)+f(x-y)=2f(x)cosy,\forall x,y \in \mathbb{R} \end{matrix}\right.$
Ta giải bt tổng quát với $f(0)=a$ , $f(\frac{\pi}{2})=b$
$P(0;y)\Rightarrow f(y)+f(-y)=2a.cos y$ $\forall y\in\mathbb{R}$
$P(x;\frac{\pi}{2})\Rightarrow f(x+\frac{\pi}{2})+f(x-\frac{\pi}{2})=0\Rightarrow f(x)+f(x+\pi)=0$ $\forall x\in\mathbb{R}$
$P(\frac{\pi}{2};y)\Rightarrow f(\frac{\pi}{2}+y)+f(\frac{\pi}{2}-y)=2b.cos y\Rightarrow f(\frac{\pi}{2}-y)-f(y-\frac{\pi}{2})=2b.cos y$ $\forall y\in\mathbb{R}$
Mà $f(\frac{\pi}{2}-y)+f(y-\frac{\pi}{2})=2a.cos (\frac{\pi}{2}-y)=2a.sin y$ $\forall y\in\mathbb{R}$
$\Rightarrow f(y-\frac{\pi}{2})=a.sin y-b.cos y$ $\forall y\in\mathbb{R}$
Thay $y$ bởi $x+\frac{\pi}{2}\Rightarrow f(x)=a.sin (x+\frac{\pi}{2})-b.cos (x+\frac{\pi}{2})=a.cos x+b.sin x$ $\forall x\in\mathbb{R}$
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