# Question #b0c52

Apr 17, 2017

You need to find the constant!

#### Explanation:

I think you have a typo: you mean $\frac{2}{9} {\left({x}^{3} + 1\right)}^{\frac{3}{2}}$.

Anyway, the trick is that $f \left(x\right)$ is AN antiderivative, so $f \left(x\right) = \frac{2}{9} {\left({x}^{3} + 1\right)}^{\frac{3}{2}} + C$ where $C$ is a real number such that $f \left(2\right) = 0$. So, in order to find $C$, we have that

$\frac{2}{9} {\left({2}^{3} + 1\right)}^{\frac{3}{2}} + C = 0 R i g h t a r r o w C = - 6$.

Hence $f \left(x\right) = \frac{2}{9} {\left({x}^{3} + 1\right)}^{\frac{3}{2}} - 6$, so $f \left(0\right) = \frac{2}{9} - 6 = - \frac{10}{9}$