Darwin observed that the finches on two different islands in the galapagos belonged to the same family, but showed a lot of variations in their beak size and shape. how can this variation be justified?

This is a classic example of speciation. The two subspecies of finches were separated by a geographical factor, in this case the sea/ water between the two islands. The environment varied between the islands e.g. one island might have had an abundance of seeds whereas the other island had very few to no seeds but an abundance of insects.

A random mutation in both finch subspecies led to the development of phenotypes such as a blunt and broader beak or thinner but sharper beaks. On the island with an abundance of seeds, those finches with a broader blunter beak had a selective advantage, as they could break through the tough seeds and eat them. This meant they outcompeted the ones with the sharper beaks. This meant they could survive, breed and pass on their alleles to their offspring. Due to natural selection, the allele frequency ( the number of broader beaks) in the population of finches on this specific island increased. This continued until the majority of the population were broad-beaked. Vice versa for the Sharp beaks on the island with the insects. 

I'm soooo sorry but i can't answer but you can ask another professor