Question

Fie functia f:(0, ∞) -> R, f(x) x+1/x (totul supra). Calculati f(1)*f(2)*...*f(2008).

Answer 1

$f(x) = \dfrac{x+1}{x}\\ \\\\ f(x)\cdot f(x+1) = \dfrac{x+1}{x}\cdot \dfrac{x+2}{x+1} = \dfrac{x+2}{x}\\ \\ f(x)\cdot f(x+1)\cdot f(x+2) = \dfrac{x+2}{x}\cdot \dfrac{x+3}{x+2} = \dfrac{x+3}{x} \\ \\ \vdots\quad \text{(analog)} \\ \\ f(x)\cdot f(x+1)\cdot f(x+2)\cdot ...\cdot f(x+n) = \dfrac{x+(n+1)}{x}\\ \\\\ \Rightarrow f(1)\cdot f(2)\cdot...\cdot f(2008) = \\\\ =f(1)\cdot f(1+1)\cdot f(1+2)\cdot ...\cdot f(1+2007) = \dfrac{1+(2007+1)}{1} =\\ \\ = \boxed{2009}$

Answer 2

Răspuns:

2/1×3/2×4/3×...×2008/2007.×2009/2008=2009/2007 ;re3stu se simplifica;f(1)=2/1 ...

Explicație pas cu pas: