In the maximum subarray problem, we are given a vector of integers vec and want to find the maximum sum of entries in a contiguous subvector. That is$,$ we want to find indices ij so that vec$[$i$]+$ vec$[$i$+1]+...+$ vec$[$j$]$ is as large as possible. Here is an example in the pictured image. In this example the maximum sum of entries in a contiguous subvector is from index $2$ to index $6$ and has sum $187.$

We can approach the maximum subarray problem using dynamic programming. Let largestSumTo be a vector where largestSumTo$[$j$]$ is the maximum over ij of vec$[$i$]+$ vec$[$i$+1]+...+$ vec$[$j$],$ i$.$e$.,$ the largest sum of contiguous entries that ends at index j$.$

We set largestSumTo$[0]=$ vec$[0]$ and then successively compute largestSumTo$[1],$ largestSumTo$[2],...,$ largestSumTo$[$n$],$ when the input vector has size n$.$ For j$>0,$ how can we express largestSumTo$[$j$]$ in terms of the previous entries of largestSumTo $?$ pick an answer

a$.$ largestSumTo$[$j$]=$ vec$[$j$]+$ largestSumTo$[$j$-1]$

b$.$ largestSumTo$[$j$]=$ max$($vec$[$j$]+$ largestSumTo$[$j$-1],$ vec$[$j$])$

c$.$largestSumTo$[$j$]=$ largestSumTo$[$j$-1]+$ largestSumTo$[$j$-2]$

d$.$ largestSumTo$[$j$]=$ max$($vec$[$j$],$ vec$[$j$]+$ vec$[$j$-1])$