Can you have an inflammation without having fever? for example , when you have a cold can u take an antibiotic like augmentin (amoxicillin and clavulanate) despite no fever ?

Inflammation: You use a poor example because antibiotics are not indicated for viruses such as colds, but in answer to your question there are occasions when you may have an inflammtion without fever especially if you are debilitated.

No antibiotics: A cold is a viral infection. Antibiotics treat only bacterial infections. Taking any antibiotic when you have a cold is a bad idea. It does absolutely nothing to treat the cold, and puts you at unnecessary risk of side effects from the antibiotic. In addition, bacteria are becoming increasingly resistant to antibiotics because of their inappropriate use to treat viral infections.