Cho hàm số f(x) nhận giá trị dương và thỏa mãn f(0)=1

Ta có: ${\left(f^{\text{'}}\left(x\right)\right)}^3=e^x{\left(f\left(x\right)\right)}^2,\forall x\in ℝ\iff f^{\text{'}}\left(x\right)=\sqrt[3]{e^x}.\sqrt[3]{{\left(f\left(x\right)\right)}^2}\iff \frac{f^{\text{'}}\left(x\right)}{\sqrt[3]{{\left(f\left(x\right)\right)}^2}}=\sqrt[3]{e^x}$.

$\Rightarrow \underset{0}{\overset{3}{\int }}\frac{f^{\text{'}}\left(x\right)}{\sqrt[3]{{\left(f\left(x\right)\right)}^2}}dx=\underset{0}{\overset{3}{\int }}\sqrt[3]{e^x}dx\iff \underset{0}{\overset{3}{\int }}\frac{1}{\sqrt[3]{{\left(f\left(x\right)\right)}^2}}df\left(x\right)=\underset{0}{\overset{3}{\int }}{e}^{\frac{x}{3}}dx\iff {\left.3\sqrt[3]{f\left(x\right)}\right|}_0^3={\left.3e^{\frac{x}{3}}\right|}_0^3$

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

Chọn đáp án B