You know that the bending stress is going to be a maximum at the point the farthest from the neutral axis. All along the neutral axis, the bending stress must equal zero.
Bending stresses are either tensile or compressive, so the combined bending stress at a particular location on the cross section can be found by adding together algebraically all of the different bending stresses which occur at that location.
To solve your problem, you must figure out, for each bending moment, which stresses are compressive, which are tensile, and where each stress if located on the circular cross section, in order to find the correct maximum bending stress produced by the combined loading on this structure.
Unlike a bar which has a rectangular or square cross section, a circular cross section can have infinitely many neutral axes, because any diameter passes thru the centroid of the section. The direction in which the bending moment is applied becomes the determining factor in locating where the maximum bending stress is located.
For instance, the moment created by applying F1 to the pole in your problem acts about the z-axis, which means the maximum stress for this moment will be found on the x-axis. The moment created by F2 causes bending to occur about the x-axis, but the maximum stresses will be found on the z-axis. Because the two bending moments have different magnitudes, the maximum combined bending stress max occur on these axes, or somewhere in between. You can be sure until you make the calculations and analyze the bending stresses further.
You absolutely cannot just calculate bending stress due to the total moment and say that gives the maximum bending stress.
