The Randić index R(G) of a graph G is defined by , where d(u) is the degree of a vertex u and the summation extends over all edges uv of G. The eccentricity ϵ(v) of a vertex v is the maximum distance from it to any other vertex and the average eccentricity cdd1c78229aad"> of graph G is the mean value of eccentricities of all vertices of G. There are relations between the Randić index and the average eccentricity of connected graphs conjectured by a computer program called AGX: for any connected graph G on n≥14 vertices, both lower bounds of and are achieved only by a star. In this paper, we show that both conjectures are true.