The continuous improvement of productivity level puts forward higher requirements for regional economic development strategies and paths. For the extreme value problem with constraints involved in the regional economic development path, this paper adopts the Lagrange multiplier method to accomplish it. Firstly, the implicit function existence theorem is utilized to explore the necessary conditions for the existence of the extreme value of multivariate functions. Then the operation principle and process of Lagrange multiplier method are described in detail, and the solution steps of Lagrange multiplier method in economic optimization problems, utility maximization and cost optimization are elaborated successively. In the application research of regional economic development path, the regression coefficient of the impact of utility maximization and cost optimization on regional economic growth is 0.2483, which passes the test of 1% significance level. By applying the Lagrange multiplier method to the study of regional economy, the rapid development of regional economy is effectively promoted.
