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Leveraging recent advances on mobile edge computing (MEC), edge intelligence has emerged as a promising paradigm to support mobile artificial intelligence (AI) applications at the network edge. In this paper, we consider the AI service placement problem in a multi-user MEC system, where the access point (AP) places the most up-to-date AI program at user devices to enable local computing/task execution at the user side. To fully utilize the stringent wireless spectrum and edge computing resources, the AP sends the AI service program to a user only when enabling local computing at the user yields a better system performance. We formulate a mixed-integer non-linear programming (MINLP) problem to minimize the total computation time and energy consumption of all users by jointly optimizing the service placement (i.e., which users to receive the program) and resource allocation (on local CPU frequencies, uplink bandwidth, and edge CPU frequency). To tackle the MINLP problem, we derive analytical expressions to calculate the optimal resource allocation decisions with low complexity. This allows us to efficiently obtain the optimal service placement solution by search-based algorithms such as meta-heuristic or greedy search algorithms. To enhance the algorithm scalability in large-sized networks, we further propose an ADMM (alternating direction method of multipliers) based method to decompose the optimization problem into parallel tractable MINLP subproblems. The ADMM method eliminates the need of searching in a high-dimensional space for service placement decisions and thus has a low computational complexity that grows linearly with the number of users. Simulation results show that the proposed algorithms perform extremely close to the optimum and significantly outperform the other representative benchmark algorithms.