Consider a file whose size varies between 4 kb and 4 mb during its lifetime. which of the three allocation schemes (contiguous, linked and table/indexed) will be most ap- propriate

Table/indexed.  
Let's look at the three options and see what their advantages and disadvantages are:  
Contiguous - In this scheme, the file is stored in contiguous blocks of the disk. It allows for easy random access of the data, but requires a contiguous sequence of blocks large enough to handle the entire file. Since the size of the file specified in this question varies quite a bit over it's lifespan, you're either going to be wasting a lot of space by having an allocation large enough to handle the maximum sized file, or the file will need to be copied whenever it grows and "bumps" into a file that was allocated after it. Because of this, this method is not the best.  
Linked - The file is stored as a single, or double linked list of file blocks. This allows for the file to grow or shrink as needed, using only the amount of space needed for the file. Unfortunately, this storage scheme doesn't allow for random access of the file contents and the file can only be accessed sequentially. The question for this problem doesn't specify how the file is being accessed, so as long as random access isn't required, then this would be a reasonable allocation scheme. But I'm assuming that random access will be required, in which case, this scheme isn't ideal.  
table/indexed - In this scheme, some disk blocks are used as tables to point to other disk blocks that actually contain the file data. It's almost as fast as contiguous allocation for random access of the file contents, yet allows for the growth and shrinkage of a file like linked allocation. As such, it handles all use cases at a relatively minor cost in total storage required. So this would be the most appropriate allocation scheme since the file access behavior wasn't specified in this question.