[PhD Session only] Let D=(V,A;c,) be a flow network with positive integer capacity vector c and positive flow price vector . We wish to maximize the s - t flow subject to the additional constraint that the flow cost does not exceed a given cost budget B. (a) Let be the max-flow value from the source s to the sink t. For each [0,], let mc() denote the the min-cost of s - t flows of value . Prove that mc() strictly increases with [0,]