Cho hàm số \(y = f\left( x \right)\) thỏa mãn \(f'\left( x \right).f\left( x \right) = {x^4} + {x^2}\). Biết \(f\left( 0 \right) = 2\). Tính \({f^2}\left( 2 \right)\)

* Hướng dẫn giải

Ta có:

\(\begin{array}{l}\,\,\,\,\,\,\,f'\left( x \right).f\left( x \right) = {x^4} + {x^2}\\ \Rightarrow \int\limits_0^2 {f'\left( x \right).f\left( x \right)dx}  = \int\limits_0^2 {\left( {{x^4} + {x^2}} \right)} dx\\ \Leftrightarrow \left. {\dfrac{1}{2}{f^2}\left( x \right)} \right|_0^2 = \left. {\left( {\dfrac{1}{5}{x^5} + \dfrac{1}{3}{x^3}} \right)} \right|_0^2\\ \Leftrightarrow \dfrac{1}{2}\left( {{f^2}\left( 2 \right) - {f^2}\left( 0 \right)} \right) = \left( {\dfrac{1}{5}.32 + \dfrac{1}{3}.8} \right) - 0\\ \Leftrightarrow {f^2}\left( 2 \right) - {2^2} = \dfrac{{272}}{{15}} \Leftrightarrow {f^2}\left( 2 \right) = \dfrac{{332}}{{15}}.\end{array}\)

Chọn B.