Let $f(x)$ be positive and continuous in $[a,\infty)$ and exists a finite limit for $\lim\limits_{x\to \infty} f(x) $.
Let $g(x) = 5f^{(3)}(x) + 2f(x)$
Prove or disprove - $\int_{a}^{\infty} f(x) dx$ converges if and only if $\int_a^\infty g(x) dx $ converges
I thought using a comparison test, but can I prove that $g(x)$ is positive?
Thank you!