This problem will require some dynamic programming. Think of how to solve it in O(n)

This is an extremely common technical interview question. Kadane's algorithm is a well known algorithm for solving this exact problem. The algorithm works by scanning through the input array and computing the maximum up to each point in the array.

largestSubarraySum(int input[])
    int maxSoFar = 0;
    int maxEndingHere = 0;

    for (int n = 0; n < input.size(); n++)
       maxEndingHere += input[n];

       if (maxEndingHere < 0)
          maxEndingHere = 0;
       else if (maxSoFar < maxEndingHere)
          maxSoFar = maxEndingHere;

    return maxSoFar;

Note: The algorithm will not work for all negative numbers. Consider using absolute values or other techniques for such problems.
Check here for more information on Kadane's algorithm.