find the index of a locally minimum value. All integers are different. A value is locally minimal if it is smaller than both of its neighbors $($or one neighbor if it is at the end of the array$).$ For instance, given array A $=[6,3,4,7,9,11],$ the number $3($located at index $2)$ is locally minimal. Prove the correctness of your algorithm and analyze its runtime. The na $$ve algorithm to beat runs in linear time. $($HINT: To prove correctness, you should also prove the existence of a local minimum.$)$