The filament material with the lowest melting point among the given options is Osmium. Let's understand why this is the case.

Melting point is defined as the temperature at which a solid material changes into a liquid state. Different materials have different melting points based on their atomic and molecular structures. In the case of filament materials, the melting point is an important property as it determines the temperature at which the filament will start to melt and lose its structural integrity.

Now, let's discuss the melting points of the given filament materials:

a) Carbon:
- Carbon has a high melting point of about 3550°C (6410°F) for diamond form and around 3500°C (6330°F) for graphite form.
- Due to its strong covalent bonds, carbon has an exceptionally high melting point, making it unsuitable for use as a filament material in typical applications.

b) Tungsten:
- Tungsten has a very high melting point of about 3422°C (6192°F).
- It has the highest melting point of all metals and is commonly used in incandescent light bulb filaments and other high-temperature applications.

c) Osmium:
- Osmium has the lowest melting point among the given options, with a melting point of about 3033°C (5491°F).
- Although Osmium has a relatively high melting point compared to many other materials, it is the lowest among the given choices.

d) Tantalum:
- Tantalum has a high melting point of about 3017°C (5463°F).
- It is commonly used in high-temperature applications, including lamp filaments and electronic components.

In summary, Osmium has the lowest melting point among the given filament materials, making it the most suitable option for applications where lower temperature stability is required. However, it is important to note that Osmium is a rare and expensive material, and therefore, alternative materials with lower melting points are often preferred in practical applications.