Cho hàm số \(f\left( x \right)\) nhận giá trị dương và thỏa mãn \(f\left( 0 \right)=1, {{\left( {f}'\left( x \right) \right)}^{3}}={{e}^{x}}{{\left( f\left( x \right) \right)}^{2}}...

* Hướng dẫn giải

\({\left( {f'\left( x \right)} \right)^3} = {e^x}{\left( {f\left( x \right)} \right)^2},\,\forall x \in R \Leftrightarrow f'\left( x \right) = \sqrt[3]{{{e^x}}}.\sqrt[3]{{{{\left( {f\left( x \right)} \right)}^2}}} \Leftrightarrow \frac{{f'\left( x \right)}}{{\sqrt[3]{{{{\left( {f\left( x \right)} \right)}^2}}}}} = \sqrt[3]{{{e^x}}}\)

\( \Rightarrow \int\limits_0^3 {\frac{{f'\left( x \right)}}{{\sqrt[3]{{{{\left( {f\left( x \right)} \right)}^2}}}}}dx} = \int\limits_0^3 {\sqrt[3]{{{e^x}}}dx} \Leftrightarrow \int\limits_0^3 {\frac{1}{{\sqrt[3]{{{{\left( {f\left( x \right)} \right)}^2}}}}}df\left( x \right)} = \int\limits_0^3 {{e^{\frac{x}{3}}}dx} \Leftrightarrow \left. {3\sqrt[3]{{f\left( x \right)}}} \right|_0^3 = \left. {3{e^{\frac{x}{3}}}} \right|_0^3\)

\(\sqrt[3]{{f\left( 3 \right)}} - \sqrt[3]{{f\left( 0 \right)}} = e - 1 \Leftrightarrow \sqrt[3]{{f\left( 3 \right)}} - 1 = e - 1 \Leftrightarrow f\left( 3 \right) = {e^3}\)