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Maximum Satisfiability problem is the central problem in theoretical computer science. Local search has been testified to be effective in the practice to solve these problems. This paper presents new local search methods to solve the maximum satisfiability problems and analyzes the performance of the methods. We focus on the sub problem with each clause containing at least k literals, Max-(k)-Sat briefly. The central issue is to discuss the local search algorithms as well as their performance to solve Max-(2)-Sat and Max-(3)-Sat. We first propose a local search algorithm to solve Max-(2)-Sat. This algorithm can achieve the performance ratio not larger than 4/3 and can be extended to solve Max-(k)-Sat with performance ratio not larger than k+2/k+1 for k ≥ 3. We then propose a local search algorithm for Max-(3)-Sat and show that the algorithm can achieve the performance ratio not larger than 8/7. This algorithm can be extended to solve Max-(k)-Sat with performance ratio not larger than 2k+2/2k+1 for k ≥ 4. We can give examples to show that the aforementioned bounds are all tight. The algorithm for Max-(3)-Sat can naturally derive the local search algorithms for Not-2Sat and Not-3Sat with performance ratio 2 and 4/3 respectively.