Answer: \(\mathrm f(x)=\frac14(3x+5)^{4/3}+4x+12\).

Set up the integral using the correct limits or antiderivative first. When definite integration is used, evaluate the upper limit and lower limit separately before subtracting.

Integrate \(\mathrm f''(x)=(3x+5)^{-2/3}\):

\(\mathrm f'(x)=(3x+5)^{1/3}+C\).

Using \(\mathrm f'(1)=6\),

\(6=8^{1/3}+C=2+C\), so \(C=4\).

Thus

\(\mathrm f'(x)=(3x+5)^{1/3}+4\).

Integrate again:

\(\mathrm f(x)=\frac14(3x+5)^{4/3}+4x+D\).

Using \(\mathrm f(1)=20\),

\(20=\frac14(8)^{4/3}+4+D\).

Since \(8^{4/3}=16\), this gives

\(20=4+4+D\), so \(D=12\).

Therefore

\(\mathrm f(x)=\frac14(3x+5)^{4/3}+4x+12\).

This gives the required answer: \(\mathrm f(x)=\frac14(3x+5)^{4/3}+4x+12\).