努める

to endeavor (endeavour) /to try/to strive/to make an effort/to exert oneself/to be diligent/to be committed (to doing something)

頑張る

(1) to persevere/to persist/to keep at it/to hang on/to hold out/to do one's best/ (2) to insist that/to stick to (one's opinion) / (3) to remain in a place/to stick to one's post/to refuse to budge

勉める

力める