Functie

[DrAGos Calinescu]

Determinati toate functiile \( f:\mathbb{R}\rightarrow\mathbb{R} \) cu urmatoarea proprietate
\( f(x+y)+f(x-y)=2f(x)\cos y \)

[Mateescu Constantin]

Sa se determine toate functiile \( f\ :\ \mathbb R\to\mathbb R \) astfel incat \( f(x+y)+f(x-y)=2f(x)\cos y \) .

Notam \( f(0)=a \) si \( f\(\frac{\pi}{2}\)=b \) , \( a \) , \( b\in\mathbb R \) . Dam succesiv valori lui \( x \) si \( y \) si obtinem :

\( \left\|\ \begin{array}{ccc}
x & = & 0 &,& y & = & k & \Longrightarrow & f(k) & + & f(-k) & = & 2a\cos k \\\\\\\\
x & = & \frac{\pi}2+k &,& y & = & \frac{\pi}2 & \Longrightarrow & f(\pi+k) & + & f(k) & = & 0 \\\\\\\\
x & = & \frac{\pi}2 &,& y & = & \frac{\pi}2+k & \Longrightarrow & -f(\pi+k) & - & f(-k) & = & 2b\sin k\ \end{array}\right|\ \bigoplus\ \Longrightarrow\ \overline{\underline{\left\|\ f(k)\ =\ a\cos k\ +\ b\sin k\ \right\|}} \)