No.
E.g., take $a=4$ and $b=27/8$. Then the positive roots of $g$ are $x_1\approx0.338$, $x_2\approx0.826$, $x_3\approx3.78$, and $f(x_1)\approx13.6$ and $f(x_3)\approx9.72$, so that $x_1$ is not a global minimizer of $f$ on $(0,\infty)$.
For an illustration, here are graphs of $f$ (blue) and $g$ (red):
