Question

If a continuous function $f$ satisfies ${\int }_0^{f\left(x\right)}{t}^{2}dt=x^2(1+x)$ for all $x\ge 0$ then $f\left(2\right)$ is equal to

• A. $12$
• B. $\sqrt[3]{336}$
• C. $3$
• D. $\sqrt[3]{342}$

Answer 1

The correct option is B $\sqrt[3]{336}$

${\int }_0^{f\left(x\right)}{t}^{2}dt=x^2(1+x)\Rightarrow \frac{{\left(f\left(x\right)\right)}^3}{3}=x^2(1+x)\Rightarrow f\left(2\right)=\sqrt[3]{33\times 4\times 3}=\sqrt[3]{336}$