 Solution

For compactly supported variations, antisymmetry of the electromagnetic field tensor gives $\delta (-F_{ab}{F}^{ab}/4)=-F^{ab}{\partial }_a\delta A_b$. Integrate by parts in the action to obtainArbitrariness of the variation gives the Proca equationtogether with the unchanged identity ${\partial }_{[a}{F}_{bc]}=0$ coming from $F=dA$. The signs follow from the source and mass terms as printed.