Answer
$$f(x)=\frac{4}{15}x^{5/2}+c_1x+c_2$$
Work Step by Step
Given $$f''(x)= \sqrt{x}$$
integrate both sides
\begin{align*}
f'(x)&=\int x^{1/2}dx\\
&=\frac{2}{3}x^{3/2}+c_1
\end{align*}
To find $f(x) $
\begin{align*}
f(x)&=\int f'(x)dx\\
&=\int \left(\frac{2}{3}x^{3/2}+c_1\right)dx\\
&=\frac{4}{15}x^{5/2}+c_1x+c_2
\end{align*}
