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Câu hỏi: Cho hàm số $f\left( x \right)$ thỏa mãn $f\left( 1 \right)=\sqrt[3]{3e}$ và ${f}'\left( x \right)=\dfrac{{{e}^{x}}}{{{\left[ f\left( x \right) \right]}^{2}}}$ $\left( {f}'\left( x \right)\ne 0 \right)$. Tìm giá trị của ${{\left[ f\left( 2 \right) \right]}^{3}}$
A. ${{e}^{2}}$.
B. ${{e}^{3}}$.
C. $3{{e}^{3}}$.
D. $\dfrac{1}{3}{{e}^{3}}$.
Ta có ${f}'\left( x \right)=\dfrac{{{e}^{x}}}{{{\left[ f\left( x \right) \right]}^{2}}}\Leftrightarrow {{\left[ f\left( x \right) \right]}^{2}}.{f}'\left( x \right)={{e}^{x}}$
$\Leftrightarrow \int{{{\left[ f\left( x \right) \right]}^{2}}.{f}'\left( x \right)\text{d}x}=\int{{{e}^{x}}\text{d}x}$ $\Leftrightarrow \dfrac{{{\left[ f\left( x \right) \right]}^{3}}}{3}={{e}^{x}}+c$ $\Leftrightarrow f\left( x \right)=\sqrt[3]{3{{e}^{x}}+3c}$
$f\left( 1 \right)=\sqrt[3]{3e}$ $\Leftrightarrow {{\left[ f\left( 1 \right) \right]}^{3}}=3e$ $\Leftrightarrow 3e=3{{e}^{1}}+3c$ $\Leftrightarrow c=0$
Vậy $f\left( x \right)=\sqrt[3]{3{{e}^{x}}}$, ${{\left[ f\left( 2 \right) \right]}^{3}}=3{{e}^{2}}$.
A. ${{e}^{2}}$.
B. ${{e}^{3}}$.
C. $3{{e}^{3}}$.
D. $\dfrac{1}{3}{{e}^{3}}$.
Ta có ${f}'\left( x \right)=\dfrac{{{e}^{x}}}{{{\left[ f\left( x \right) \right]}^{2}}}\Leftrightarrow {{\left[ f\left( x \right) \right]}^{2}}.{f}'\left( x \right)={{e}^{x}}$
$\Leftrightarrow \int{{{\left[ f\left( x \right) \right]}^{2}}.{f}'\left( x \right)\text{d}x}=\int{{{e}^{x}}\text{d}x}$ $\Leftrightarrow \dfrac{{{\left[ f\left( x \right) \right]}^{3}}}{3}={{e}^{x}}+c$ $\Leftrightarrow f\left( x \right)=\sqrt[3]{3{{e}^{x}}+3c}$
$f\left( 1 \right)=\sqrt[3]{3e}$ $\Leftrightarrow {{\left[ f\left( 1 \right) \right]}^{3}}=3e$ $\Leftrightarrow 3e=3{{e}^{1}}+3c$ $\Leftrightarrow c=0$
Vậy $f\left( x \right)=\sqrt[3]{3{{e}^{x}}}$, ${{\left[ f\left( 2 \right) \right]}^{3}}=3{{e}^{2}}$.
Đáp án C.