Answer
$$f(x) = \frac{3}{5}x^{5/3} $$
Work Step by Step
Given $$f^{\prime}(x)=x^{2 / 3}$$
Since
\begin{align*}
f(x)&=\int f'(x)dx\\
&= \int x^{2/3}dx\\
&= \frac{3}{5}x^{5/3}+C
\end{align*}
Since $f(3/5)= 1$, then $C= 0$ and
$$f(x) = \frac{3}{5}x^{5/3} $$