Solve stock, BFS path, and merge intervals

Solve three coding problems; justify complexity and corner cases. A) Best Time to Buy/Sell Stock (one transaction): Given prices[0..n-1] (integers), return (max_profit, buy_day, sell_day). If no profitable trade exists, return (0, -1, -1). Achieve O(n) time, O(1) space. Prove correctness on strictly decreasing arrays and when multiple equal minima/maxima exist; prefer earliest buy when profits tie. B) Shortest path with forbidden nodes: Given an unweighted directed graph G with nodes 1..N (N ≤ 1e5), adjacency list, source s, target t, and a forbidden set F, return the lexicographically smallest shortest path from s to t that avoids F, or [] if none. Use BFS. Explain how to achieve O(N+M) time without sorting neighbors on every expansion, and how you’d adapt if the input edges arrive unsorted. C) Merge half-open intervals: Given intervals [li, ri) with 32-bit integers, merge any that overlap or just touch (e.g., [1,3) and [3,5) ⇒ [1,5)). Return a minimal, start-sorted set. Handle negatives and extremes without overflow, and prove that the total covered measure equals the sum of merged lengths. Provide tests for edge cases.