||Mean Value Theorems in Higher Dimensions and Their Applications
||Wei-Chi YANG and Vladimir SHELOMOVSKII
In this paper, we describe the Mean Value Theorem (MVT) and Cauchy Mean Value Theorem (CMVT) when considering an ℝn-1 dimensional hyperplane intersects an ℝn-1 dimensional smooth surface in ℝn. We demonstrate how we derive the the proofs of MVT and CMVT by applying techniques described in . We further discuss how the theorems can be extended by replacing the hyperplane with another smooth surface. Next, we link MVT to problems of ?nding the extreme values for a smooth function subject to several constraints. We use technological tools to show how we can obtain the solutions that are guaranteed by our theories.