Đáp án

\(\frac{[f(x)]^2 [f'(x)]^2}{e^{2x}} = 1 + (f(x))^2\)  

\(\Rightarrow \frac{f'(x) f(x)}{e^x} = \sqrt{1 + (f(x))^2} \Rightarrow \frac{f'(x) f(x)}{\sqrt{1 + (f(x))^2}} = e^x\)  

\(\Rightarrow  \int \frac{f'(x) f(x)}{\sqrt{1 + (f(x))^2}} dx = \int e^x dx\)  

\(\Rightarrow \sqrt{1 + (f(x))^2} = e^x + C.\)  

\(f(0) = 1 \Rightarrow C = \sqrt{2} - 1\)  

\(\Rightarrow \sqrt{1 + (f(x))^2} = e + (\sqrt{2} - 1)\)  

\(\Rightarrow f(1)^2 = (e+ \sqrt{2} - 1)^2 - 1\)  

\(\Rightarrow f(1) = \sqrt{(e + \sqrt{2} - 1)^2 - 1} \approx 2,96 \Rightarrow \boxed{A}\)